WPS3642
ADVANCES IN NEGOTIATION THEORY:
BARGAINING, COALITIONS, AND FAIRNESS
Carlo Carraro, Carmen Marchiori and Alessandra Sgobbi*
Abstract
Bargaining is ubiquitous in real life. It is a major dimension of political and
business activities. It appears at the international level, when governments
negotiate on matters ranging from economic issues (such as the removal of
trade barriers), to global security (such as fighting against terrorism) to
environmental and related issues (e.g. climate change control). What factors
determine the outcomes of negotiations? What strategies can help reach an
agreement? How should the parties involved divide the gains from
cooperation? With whom will one make alliances? This paper addresses these
questions by focusing on a Noncooperative approach to negotiations, which is
particularly relevant for the study of international negotiations. By reviewing
Noncooperative bargaining theory, Noncooperative coalition theory, and the
theory of fair division, this paper will try to identify the connections among
these different facets of the same problem in an attempt to facilitate progress
toward a unified framework.
World Bank Policy Research Working Paper 3642, June 2005
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage
the exchange of ideas about development issues. An objective of the series is to get the findings out
quickly, even if the presentations are less than fully polished. The papers carry the names of the authors
and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper
are entirely those of the authors. They do not necessarily represent the view of the World Bank, its
Executive Directors, or the countries they represent. Policy Research Working Papers are available
online at http://econ.worldbank.org.
This paper is a product of the study "Stable arrangements for allocation of water among competing uses
under stochastic supply conditions". The study is funded by DECRG. The study team includes: Carlo
Carraro, Carmen Marchiori, Irene Parachino, Fioravante Patrone, Alessandra Sgobbi, Stefano Zara, and
Ariel Dinar (TTL).
*Carlo Carraro is Professor of Environmental Economics at the University of Venice, Department of
Economics, San Giobbe 873, 30121, Venice, Italy, and Research Director of the Fondazione Eni E.
Mattei. Carmen Marchiori is a Ph.D. Student at the London School of Economics. Alessandra Sgobbi is
junior researcher at the Fondazione Eni E. Mattei. The authors are grateful to Ariel Dinar, Zmarak Shalizi
and to Anastasios Xepapadeas for helpful comments on a previous version of this paper.
Table of Contents
1 Introduction _____________________________________________________________ 1
2 Cooperative versus Noncooperative Bargaining theory __________________________ 3
3 Fundamentals of Noncooperative Bargaining Theory. The basic Rubinstein
Alternating-Offer Game ________________________________________________________ 6
3.1 Structure of the game ________________________________________________ 6
3.2 Assumptions on players' preferences____________________________________ 8
3.3 Main Results________________________________________________________ 9
4 Extensions of the Standard Noncooperative Bargaining Model___________________ 11
4.1 Multiple Players ____________________________________________________ 11
4.2 Multiple Issues _____________________________________________________ 14
4.3 Incomplete Information _____________________________________________ 18
4.4 Bargaining in stochastic environments _________________________________ 20
4.5 Repeated Bargaining Situations_______________________________________ 23
4.6 Synthesis of the results ______________________________________________ 25
5 Noncooperative Coalition Theory___________________________________________ 27
5.1 Cooperative versus Noncooperative coalition theory______________________ 28
5.2 Simultaneous (Noncooperative) games _________________________________ 30
5.3 Sequential (Noncooperative) games____________________________________ 33
5.4 Coalition Formation and Negotiations__________________________________ 34
6 Fair-division theory ______________________________________________________ 36
6.1 Experimental Evidence ______________________________________________ 37
6.2 Theories of fair behaviour____________________________________________ 38
6.3 Fair Division Procedures_____________________________________________ 39
6.3.1 Basic allocation procedures _______________________________________ 41
6.3.2 Refinements of the basic procedures ________________________________ 42
6.4. Synthesis of the procedures______________________________________________ 45
7 Conclusions ____________________________________________________________ 48
8 References _____________________________________________________________ 49
1 Introduction
Bargaining is ubiquitous in real life. In the arena of social interaction, for example, a
married couple is almost constantly involved in negotiation processes throughout the
relationship, from the decision of who will look after the children, to the question of
whether to buy a house, how to manage the resources of the family and so on. In the
political arena, a bargaining situation exists, for example, when no single political party
on its own can form a government, but different parties have to make alliances and
agree on a common program for them to have the chance of winning. At the
international level, governments are often engaged in a variety of negotiations on
matters ranging from economic issues (such as the removal of trade barriers), to global
security (such as fighting against terrorism) to environmental and related issues (such as
a pollutant's emissions reduction, water resource management, biodiversity
conservation, climate change control, etc.).
What factors determine the outcome of negotiations such as those mentioned
above? What strategies can help reach an agreement? How should the parties involved
divide the gains from cooperation? With whom will one make alliances?
The study of any bargaining process is extremely hard, involving a multiplicity
of questions and complex issues. As a consequence, the research literature in this field
has not yet been able to develop a comprehensive framework for analysis, and a number
of theories have been proposed instead, each focusing on single aspects of the problem.
So, for instance, the issue of how to divide the payoffs from cooperation among
the parties is traditionally addressed within cooperative bargaining theory, which
makes, in turn, "beneficial" assumptions about which properties the equilibrium
allocation should have, and does not explicitly address the question of which strategies
will be adopted by the negotiators.
In many real life situations, however, cooperation cannot be ensured, and
binding agreements are not a feasible option. Therefore, the strategic choices of the
actors involved in the bargaining process need to be explicitly modeled in order to
determine the final outcome of the negotiation. Noncooperative bargaining theory is
more concerned with these situations and focuses on the bargaining procedures in the
1
attempt to determine which equilibrium outcome will prevail in the absence of
interventions.
When multiple players are involved in the bargaining, there is the possibility that
coalitions form. Traditional bargaining theory is not suitable for representing such
situations because it is based on the assumption that only two possible outcomes can
arise: the fully cooperative outcome and the fully Noncooperative outcome, where
respectively an agreement among all parties is reached and no agreement forms.
Noncooperative coalition theory considers this interesting aspect of negotiation
processes and, without making any assumption on the final result, analyzes the
incentives that players may have to form coalitions, and how the incentives may affect
the final outcome of the negotiation. The study of coalition formation is particularly
important in bargaining contexts where positive externalities are present. In this case,
due to players' incentive to free ride, it is quite unlikely that a `grand coalition' will
form; instead `partial agreements' usually arise.
Finally, traditional models of negotiation have focused almost exclusively on the
efficiency properties of both the process and the outcomes. Yet, as every day experience
indicates, considerations other than efficiency play a crucial role in selecting which
agreement will be reached if any at all and through which path. The theory of fair
division focuses on processes and strategies that respond not only to Pareto efficiency,
but also to equity, envy-freeness, and invulnerability to strategic manipulation.
Although the theoretical literature offers this classification into different
approaches to the same problem, in the applications the division is not so clear-cut. The
lack of a unified theoretical framework to address negotiations has meant that the
various isolated parts of the theory have been of little empirical use.
While recognizing the importance of cooperative game theory, this paper will
mainly focus on a Noncooperative approach to negotiations, which is particularly
relevant for the study of international negotiations. In particular, by reviewing
Noncooperative bargaining theory, Noncooperative coalition theory and the theory of
fair division, this survey will try to identify the connections among these different facets
of the same problem in an attempt to facilitate the progress towards a unified
framework.
2
The structure of the paper is as follows. Section 2 will briefly describe the
principles of cooperative bargaining theory, which represent the origins of formal
bargaining theory, and will discuss the links between the cooperative and
Noncooperative approaches in order to introduce and motivate the study of
Noncooperative bargaining. Section 3 will analyze in detail the famous alternating-offer
game proposed by Rubinstein in 1982, which constitutes the starting point for
Noncooperative bargaining theory. Section 4 will then discuss some important
extensions of this model. Section 5 will be devoted to Noncooperative coalition theory,
with the intent of providing insights into its latest developments, which seek to link the
theory of coalition formation to the bargaining process. Section 6 will be concerned
with the theory of fair division. In particular, the questions will be addressed of how
fairness considerations can alter the results of the standard theoretical models, and how
fair division algorithms can be incorporated in the existing theories. Finally, some
concluding comments will be provided in Section 7.
2 Cooperative versus Noncooperative Bargaining Theory
The formal theory of bargaining originated in the early 1950s with John Nash's work,
which establishes the basic framework of the `axiomatic (or cooperative) approach' to
negotiations.
Following Nash, a `bargaining situation' can be defined as a situation in which
(i) individuals (or "players") have the possibility of concluding a mutually beneficial
agreement, (ii) there is a conflict of interests about which agreement to conclude, and
(iii) no agreement may be imposed on any individual without his approval.
More precisely, Nash defines a `bargaining problem' to be the set of utility pairs
that can be derived from possible agreements, together with a pair of utilities which is
designated to be the `disagreement point'. This idea can be exemplified with the help of
a diagram. Figure 1 depicts a bargaining situation in which two players (A and B),
whose utilities are measured along the x and y-axis respectively, bargain over the
partition of a single cake of known size. The point denoted (dA, dB) is the point of no
agreement and it determines the minimum level of utility each party is ready to accept.
All points to the northeast of (dA, dB) represent an improvement for both players and,
3
together, they define the negotiation set1. No agreement above the frontier is feasible,
and all points on the frontier to the northeast of the no-agreement point are Pareto
efficient (that is, no player can be made better off without the other player being made
worse off by moving away from such a point).
Figure 1: The bargaining problem zone of agreement conceptualization
Utility of player B Negotiation
set
Optimal
(dA,dB) Pareto set
Utility of player A
A `bargaining solution' is a function (or formula) that assigns a single outcome
to every such problem. The Nash bargaining solution is derived from a number of
axioms about the properties that it would seem natural for the negotiation outcome to
have.
In particular, Nash proposes that a bargaining solution should satisfy the
following four axioms:
Ax1: Scale Invariance, that is, monotone transformation of the utility functions should
not alter the bargaining solution.
Ax2: Symmetry: players are identical, and so they are interchangeable. All differences
should be taken care of in the definition of the bargaining set and disagreement
points.
Ax3: Independence of irrelevant alternatives, that is, the exclusion of non-selected
alternatives from the bargaining set should not alter the bargaining solution.
1Or zone of agreement, or bargaining set.
4
Ax4: Pareto efficiency: the solution should be Pareto efficient.
It turns out that there is precisely one bargaining solution satisfying these four
axioms, and this solution has a very simple functional form: it selects the utility pair that
maximizes the product of the players' gains in utility over the disagreement outcome.
max (uA - dA)(uB - dB)
uA ,uB
Having axiomatically identified the equilibrium solution, cooperative bargaining
theory then concentrates on the problem of how to divide the benefits from agreement
among the negotiating parties. 2
A limit of this approach is that it does not capture the details of the bargaining
process. In other words, the process required to arrive at the final outcome is left un-
modeled. The justification for this is that rational actors will always choose the outcome
that maximizes their value. The most efficient solution, therefore, will always be
realized regardless of the process.
In fact, as pointed out in the introduction, in many real-life situations
cooperation cannot be ensured and binding agreements are not a feasible option because
of the absence of a legitimate authority which can impose a centralized solution and/or
the complexity of the bargaining situation often involving many parties with very
different interests. In such contexts, the strategic choices of the actors involved in the
bargaining process need to be explicitly modeled in order to determine the final
outcome of the negotiation.
Noncooperative bargaining theory, which is the focus of this review, is more
concerned with these situations and analyzes exactly the bargaining procedures, in the
attempt to find theoretical predictions of what agreement, if any, will be reached by the
bargainers. In particular, this approach seeks to identify the strategies that may sustain
cooperation and the variables that may influence agents' behaviour, such as bargaining
power, incomplete information, and power relations. In the next two sections, the
2The research literature on cooperative bargaining theory have proposed different approaches and
solutions to the basic problem analyzed by Nash. Several extensions have also been developed. See
Patrone et al. (2004) for a recent survey.
5
fundamentals of Noncooperative bargaining theory and its main extensions will be
analyzed and discussed in detail.
3 Fundamentals of Noncooperative Bargaining Theory. The Basic
Rubinstein Alternating-Offer Game
The seminal paper by Rubinstein (1982) represents the starting point for formal
Noncooperative bargaining theory. The model developed in this work proposes an
attractive and intuitive process of bargaining, and provides a basic framework which
can be adapted to many economic and non-economic situations. Sections 3.1 to 3.3 will
present and discuss the general structure of the game and its main results, while section
4 will analyze some important extensions of the model.
3.1 Structure of the game
The situation modeled by Rubinstein is the following. There are two players i = 1,2
who bargain over a single `pie' of size 1. An agreement is defined as a pair (x1, x2),
where xi is Player i's share of the pie, and the set of possible agreement is:
X = (x1, x2) R2 : x1 + x2 = 1 and xi 0 for i = 1,2 .
{ }
The players' preferences over X are diametrically opposed. Each player is
concerned only about the share of the pie that he receives, and prefers to receive more
rather than less. That is, for i=1,2 player i prefers x = (x1, x2) X to y = (y1, y2 ) X
if and only if xi > yi .
The bargaining procedure is as follows. Players can take actions only at times in
the (infinite) set T = {0,1, 2,.....}. In each periodt T one of the players, say 1, proposes
an agreement and the other player (2) either accepts the offer or rejects it. If the offer is
accepted, then the bargaining ends, and the agreement is implemented. If the proposal is
rejected, then the play passes to period (t+1), where Player 2 proposes an agreement
and Player 1 in turn accepts or rejects. The game continues in this way indefinitely until
6
an offer is accepted. At all times, each player knows all his previous moves and all
those of her opponent, then a complete information scenario is assumed.
The first two periods of the game are shown in
Figure 2. Play begins at the top of the tree, and time starts at period 0. Player 1 is
the first to move and he has a continuum of choices which corresponds to the
agreements (members of X) he can propose. Each possible proposal leads to a decision
node for Player 2, at which she accepts (A) or rejects (R) the offer. If Player 2 agrees
(right-hand branch), then the game ends and the agreement x=(x1, x2) is reached at time
t=0. If Player 2 rejects the offer (left-hand branch), then play passes to period 1, when it
is Player 2's turn to make a proposal. A typical offer of Player 2 is y=(y1, y2); for each
such offer Player 1 says A(ccept) or R(eject). If he chooses A, the game ends with the
outcome y at t=1; if he chooses R, then the game continues, Player 1 makes a further
offer, Player 2 responds, and so on.
Figure 2: The first two periods of the basic Rubinstein's alternating-offer game
Player 1
t=0
x
Player 2
A
R
x=(x1, x2)
Player 2
y t=1
Player 1
R A
y=(y1, y2)
7
3.2 Assumptions on players' preferences
To complete the description of the game, we need to specify a number of assumptions.
Rubinstein assumes that each player i=1,2 has a complete transitive reflexive preference
ordering fi over the set (X x T) U {D}3 of outcomes and that the players' preference
orderings satisfy the following conditions:
A1. (Disagreement is the worst outcome). For every (x = (x1, x2),t) X ×T and
i=1,2, we have (x = (x1, x2),t) i D = (d1,d2) .
A2. (Pie is desirable). For any t T, x X, y X and i=1,2 we have
(x = (x1, x2),t) fi (y = (y1, y2),t) if and only if xi > yi .
A3. (Time is valuable). For anyt T, s T, x X and i=1,2 we have
(x = (x1, x2),t) fi (x = (x1, x2),s) if t < s (and xi > 0 ).
A4. (Continuity). Player i's preference ordering is continuous.
A.5. (Stationarity). For any t T, x X, y X and i=1,2we have
( x = (x1, x2), t) fi (y = (y1, y2), t +1) if and only if
(x = (x1, x2), 0) fi (y = (y1, y2),1) .
A6. (Increasing loss to delay). The difference vi(xi)|t=1, where vi(xi)|t=1 is the
`present value' of (xi, t=1) for player i, is an increasing function of xi.
The first assumption concerns the `disagreement point', D, and requires that this
is the least-preferred outcome for both players. The remaining conditions concern the
behaviour of preferences on the space X x T. First of all, it is required that among
agreements reached in the same period, Player i prefers larger value of xi (A2) and
prefers to obtain any given partition of the cake sooner rather than later (A3).
Assumption A5 is then introduced in order to simplify the structure of preferences. It
requires, indeed, that the preferences between (x=(x1,x2), t) and (y=(y1,y2), s) depend
only on x, y, and the difference s t. The final condition, A6, states that the loss to delay
associated with any given amount is an increasing function of the amount.
8
An example of utility function for which conditions A1 through A6 are satisfied
is the following
Ui (xi,t) = i xi ,where i (0,1)is player i's discount factor4. The preferences
t
represented by this utility function are traditionally called time preferences with a
constant discount rate.
3.3 Main Results
The Equilibrium of the game
Rubinstein (1982) proves that every bargaining game of alternating offers in which
players' preferences satisfy A1 through A6 has a unique subgame perfect equilibrium
(SPE)5. In correspondence to this equilibrium:
Player 1 proposes the agreement x*=(x1 , x2 ), defined in equation (2.1)
* *
below, whenever it is his turn to make an offer, and accepts an offer y=(y1,
y2) of Player 2 if and only if y1 y1 ; *
Player 2 always proposes y*=(y1 , y2 ), whenever it is her turn to make an
* *
offer, and accepts an offer x=(x1, x2) of Player 1 if and only if x2 x2 . *
The outcome is that Player 1 proposes x*=(x1 , x2 ) in period 0, and Player 2
* *
immediately accepts this offer.
In particular, the SPE of the game corresponds to the unique solution of the
following equations:
y1 = v1(x1 )|t=1
* * and x2 = v2(y2 )|t=1
* * (2.1)
where the functions v1(x1 )|t=1 and v2(y2 )|t=1 represent respectively the present value of
* *
(x1 , t=1) for Player 1 and the present value of (y2 , t=1) for Player 2.
* *
In the case of time preferences with constant discount rates (i.e. Player i's
preferences over outcomes (x=(x1,x2), t) are represented by the utility function
Ui (xi,t) = i xi ), (2.1) implies that y1 =1x1 and x2 =2y2 , so that
t * * * *
3D represents the disagreement point.
4More precisely, we have i = exp(-rit), where ri is player i's discount rate and ri > 0. Therefore, if the
discount rate r decreases, then the discount factor increases. This means that player i cares more about
the future and therefore becomes more patient.
5A strategy pair is a subgame perfect equilibrium (SPE) of a bargaining game of alternating offers if the
strategy pair it induces in every subgame is a Nash equilibrium of that subgame.
9
x* = 1-
1-2 ,
12 1-122(1-1) and y* = 1(1-2), 1-1 (2.2)
1-12 1-12
Thus, if 1= 2= (that is, if the discount factors are equal), then
x*=(x1 ,x2 )=1+ ,
* * 1
1+ .
It is important to notice that as 1 approaches 1, the agreement x*=(x1 ,
*
x2 ),approaches (1,0). In other words, as Player 1 becomes more patient, his share
*
increases, and, in the limit, he receives all the pie. Similarly, as Player 2 becomes more
patient, Player 1's share of the pie approaches zero.
Properties of the equilibrium solution
The equilibrium outcome defined above displays some important properties:
P1. (Uniqueness). The SPE of the game is unique, which means that the game
has a determined solution.
P2. (No delay). Whilst the structure of the bargaining game allows negotiation to
continue indefinitely, in the unique SPE agreement is reached at time t=0.
P3. (Efficiency). From an economic point of view, the fact that negotiation ends
immediately implies that the equilibrium is efficient, in the sense that no
resources are lost in delay.
P4. (Patience). The model predicts that when a player's discount factor
increases, which means that he/she becomes more patient6, his/her negotiated
share of the pie increases. Thus, the bargaining power depends on players'
relative degree of impatience.
P5. (A-symmetry). The structure of the alternating-offer bargaining game
proposed by Rubinstein is asymmetric in one respect: one of the bargainers is the
first to make an offer. This results in an advantage for the first mover who
obtains, in the unique SPE, more than half of the pie. The asymmetry in the
structure of the game is, however, artificial and its effects can be diminished by
10
reducing the amount of time that elapses between periods. Rubinstein proves
that, in the limiting case (i.e. when the length of the periods shrinks to 0), the
amount received by a player is the same regardless of which player makes the
first offer. The unique SPE then approximates the (symmetric) Nash bargaining
solution.
4 Extensions of the Standard Noncooperative Bargaining Model
4.1 Multiple Players
Starting from the basic Rubinstein alternating-offer game described in the previous
section, most of the literature on Noncooperative bargaining theory has been devoted to
models of two players. In many real-life situations, however, bargaining processes
involve a large number of individuals or interest groups. In such a case, the prediction
of the standard model that a unique equilibrium exists where agreement is reached
immediately, does not usually hold. This section will discuss how the standard results
change in a multilateral negotiation context, which problems may arise and what
solutions have been proposed in the literature.
To simplify the discussion we consider a situation in which three players
negotiate on the partition of a cake of size 1. There are, in fact, many ways of extending
the Rubinstein two-players alternating-offer game to this case. An extension that
appears to be quite natural is the one suggested and analyzed by Shaked (1986).
Shaked's game is the following. In the first period, Player 1 proposes a partition x=(x1,
x2 , x3), with x1+x2+x3=1 and Players 2 and 3 in turn accept or reject this proposal. If
either of them rejects it, then play passes to the next period, in which it is Player 2's turn
to propose a partition and Players 3 and 1 respond sequentially. If at least one of them
rejects the proposal, then again play passes to the next period, in which Player 3 makes
an offer and Players 1 and 2 respond. Players rotate in this way until a proposal is
accepted by all responders. Players' preferences are represented by the utility function ui
= xi (where 01 is the common discount factor) and thus satisfy the assumptions
t-1
6See footnote 3.
11
A1 through A6 of the basic Rubinstein's game. Moreover, there are no exogenously
imposed limits on the duration of the game, but the absence of agreement (that is
bargaining forever) leads to a payoff of 0 for all players.
This model, of course, reduces to the standard alternating-offer game when there
are exactly two players. Unfortunately, however, for n3 the game admits many
equilibrium outcomes. In particular, it has been proved that: every allocation of the cake
can be sustained as a subgame perfect equilibrium (SPE) if players are sufficiently
patient ( >1/2) and outcomes with delay are also possible equilibria. Changing the
order of moves, the simultaneity of responses, etc., does not alter this conclusion.
The indeterminacy of the three (or n) player game has aroused much interest
among researchers, and various solutions have been proposed to isolate a unique
equilibrium outcome. Some authors, for example, suggested the adoption of different
(more refined) equilibrium concepts, while others to modify the structure of the game.
Remaining in the context of the original unanimity model introduced by Shaked,
it has been noticed that the only subgame perfect equilibrium in which players'
strategies are stationary has a form similar to the unique SPE of the two-player game. In
particular, Herrero (1985) showed that if players have time preferences with a common
constant discount factor , then this equilibrium leads to the following division of the
pie:
1+ 1 2
+ 1+ + 1+ +
, ,
2 2 2
which tends to the equal split as tends to 1.
The notion of stationary SPE may therefore be used to restore the uniqueness of
the equilibrium in multilateral bargaining situations. However, the restriction to
stationary strategies is quite strong. Such strategy prescribes actions in every period that
do not depend on time, or on events in previous periods. Thus, for example, a stationary
strategy in which Player 1 always makes the proposal (1/2, 1/2) means that even after
Player 1 has made the offer (3/4, 1/4) for a thousand times, Player 2 still believes that
Player 1 will make the offer (1/2, 1/2) in the next period, which is quite unrealistic.
A more appealing way to solve the problem of indeterminacy of the n-player
game is to modify the structure of the game. For example, Jun (1987) and Chae and
Yang (1988, 1994) consider procedures where players are engaged in a series of
12
bilateral negotiations and any player that reaches a satisfactory agreement may "exit"
the game. A more interesting approach is suggested by Krishna and Serrano (1996),
where players still have the possibility to exit, i.e. to leave with their share before the
entire bargaining process is completed, but, unlike Jun/Chae and Yang's mechanisms,
the offers are made to all players simultaneously and thus the bargaining is multilateral.
In particular, the structure of the game is as follows. There are three players
bargaining on the partition of a `pie' of size one. In the first period, Player 1 proposes a
division x=(x1, x2 , x3) and in response of such proposal the following situations can
occur: (a) Both Players 2 and 3 accept the offer. In this case, the game ends with that
division. (b) Both players reject the offer and the game passes to the next period where
Player 2 makes an offer and Players 3 and 1 must respond. (c) One of the responders,
say Player 3, accepts the offer x while the other (Player 2) rejects. In this case, 3 can
"exit" the game with an amount x3 while players 1 and 2 are left to bargain over the
division of 1 x3 in period 2. The bargaining now proceeds as in the two player
alternating offer game with player 2 proposing some partition of 1 x3 . In this model,
then, the person making offer receives a payoff if and only if all the other players accept
her offer, but a responder who is satisfied with her share, can simply `take the money
and run away', with no need for unanimous consensus as required in Shaked's game.
With the introduction of such procedure, the authors are able to identify a unique perfect
equilibrium, for any number of players. Moreover, for all n, the unique equilibrium is
characterized exactly as in the case of two players and the equilibrium agreement
approximates the n-player Nash bargaining solution when players are patient.
In our discussion of multilateral bargaining situations we have deliberately
omitted an important element that may appear in negotiation contexts with 3 or more
players, that is the possibility for players to form coalitions. This element makes the
modelling of such situations even more difficult because one should not only determine
what each player gets individually, but also which coalition will or will not form. The
study of coalition formation becomes particularly important when we consider
negotiations over public goods, such as many international environmental negotiations.
In this case, the presence of externalities may induce players to free ride on the
negotiating agreement in order to enjoy the benefits from cooperation without paying
13
any cost. These and other problems will be discussed in Section 4 which is entirely
devoted to coalition theory.
4.2 Multiple Issues
Many real-life negotiations (such as trade or environmental negotiations) do not only
involve a large number of individuals, but also a set of different issues. By contrast,
most of the existing literature focuses on the problem of dividing a `single-pie' between
two agents. In this section, we will first discuss when the insights from the classical
theory still apply to the multiple-issue case, and we will then consider other important
elements that may emerge when players negotiate over more than one project.
In general, we can distinguish two different ways of handling multiple-issue
negotiations. The first one is to bundle all the issues and discuss them simultaneously
(complete package approach7); the second one is to negotiate the issues one by one
(sequential approach). Suppose, for example, that there are two players, 1 and 2,
negotiating, via an offer-counteroffer bargaining procedure, over two different projects,
X and Y. According to the first approach, an offer is a pair (x, y) specifying a division on
both issues, and players make offers and counteroffers of (x, y) until an agreement is
reached. On the contrary, the second approach involves a sequential determination of
allocations for the two projects. For example, players may start making offers and
counteroffers on x only, until agreement. Once an agreement is reached, the allocation is
implemented and bargaining proceeds over y. Intuitively, when the first approach is
adopted, that is all issues are bargained simultaneously and allocations are implemented
only after agreement has been reached on the whole package, then even complex
negotiations reduce to "as if" single-pie bargaining and the classical theory applies
directly. This conclusion is not obvious anymore when bargaining or agreement
implementation take place according to the second approach, that is in a sequential way.
In such a case, the order in which problems are discussed may assume a strategic role
and affect the final outcome of the negotiation (in the example described above, players
could start negotiating on y instead of x and thus obtain a different result).
7In other literatures, this is known as `issue linkage'. The basic idea behind this mechanism is to design a
negotiation framework in which players do not negotiated only on one issue, but force themselves to
bargain on two or more issues jointly.
14
In multiple-issue negotiations, the timing of projects on the bargaining table is
specified by a negotiation agenda, which can be defined exogenously, i.e. before
negotiation begins, or endogenously, i.e. during the bargaining process.
In general, players may have different preferences over different agendas. In the
initial example, for instance, player 1 may prefer the agenda XY to the agenda YX, while
player 2 may prefer YX to XY. This is because: a) Players may have different time
constraints for reaching agreements on the two issues, that is each player may have its
own deadline for each issue; b) Players may differ for their attitude toward time, i.e. for
their discount factors. One player may, for instance, gain utility with time and have an
incentive to reach a late agreement (patient player), while the other may lose utility with
time and try to reach an early agreement (impatient player). In an `issue-by-issue'
bargaining process, this disposition of negotiators may strongly influence strategic
behaviors and, therefore, the negotiation outcome.
In the past decades, the study of the role of the negotiation agenda has obtained
increasing attention among researchers and various interesting contributions have been
proposed in the literature. Fershtman (1990), for example, considers a situation in which
two players with time preferences and additively separable utility functions negotiate,
according to an alternating offer procedure, over two linear issues. In this model, the
agenda is defined exogenously and both players are assumed to have identical discount
factors and no deadlines. The author analyzes sequential agendas where the realization
of utilities is postponed until both projects are accepted (simultaneous implementation).
He shows that a player prefers the first project be least important to him but most
important to the opponent. However, as players become increasingly patient, the impact
of the agenda disappears. In and Serrano (2003) develop a model to investigate exactly
the effects of agenda restrictions on the properties of the equilibrium outcome. What is
found is that when the agenda is very restricted (such as, for example, when bargainers
are forced to negotiate only one issue at a time, the one chosen by the proposer at each
round), multiple equilibria and delay in agreement do usually arise.
In a similar setting with two linear issues and two players, Busch and Horstmann
(1997) partially `endogenize' the bargaining agenda by introducing a separate
bargaining round over it. The order in which issues are negotiated becomes, however,
15
truly endogenous in Inderst (2000), where players bargain over projects without any ex-
ante agreed agenda.
In this model issues are either mutually beneficial or strictly controversial and
each subset of projects is immediately implemented after partial agreement on this set
(sequential implementation). The author first derives the equilibrium payoffs when an
exogenously given agenda requires that bargaining proceeds simultaneously or
sequentially over the set of projects. The analysis reveals that the agenda can have a
marked impact on payoffs and in contrast to the result reported by Fershtman (1990)
this impact does not seem to vanish as players become increasingly patient. In
particular, bargaining simultaneously over a set of projects can improve efficiency by
creating trading opportunities across issues8. Moreover, changing the agenda may have
a distributive effect, and players may therefore prefer different agendas. In the second
part of the paper the author then identifies which agenda is chosen endogenously. The
results of the analysis can be summarized as follows: A) when issues are mutually
beneficial, then players choose to bargain simultaneously over all issues. However, if
the bargaining set contains B) strictly controversial projects two different sub-cases
need to be distinguished depending on whether or not randomization devise is an
available option: (B1) if players have access to a randomization device, an analogous
result holds as in the previous case; (B2) with strictly controversial projects and without
lotteries there might be multiple equilibria involving even considerable delay.
8 The profitability and effectiveness of linkage strategies have been largely studied, especially in the
literature on coalition formation. Pioneering contributions are those by Tollison and Willett (1979) and
Sebenius (1983), who proposed linkage mechanisms to promote cooperation on a number of matters, such
as security and international finance. Issue linkage was formally introduced into the economic literature
on international environmental cooperation by Folmer et al. (1993) and by Cesar and De Zeeuw (1996) to
solve the problem of asymmetries among countries. The intuition is that, if some countries gain from
cooperating on a given issue whereas other countries gain from cooperating on another one, by linking the
two issues it may be possible to obtain an agreement which is profitable to all countries. Linkage
strategies can also be used to mitigate the problem of free-riding which normally affect negotiations over
public goods, such as environmental quality. This aspect has been addressed in various ways. For
instance, Barrett (1995, 1997), proposes linking environmental protection to negotiations on trade
liberalisation. In this way, potential free-riders are deterred with threats of trade sanctions. Other
interesting contributions are those by Carraro and Siniscalco (1995, 1997), and Katsoulacos (1997),
where environmental cooperation is linked to cooperation in R&D. In a more recent work, Alesina et al.
(2001) further analyze the problem of the effectiveness of linkage mechanisms in increasing cooperation,
and identify an interesting trade-off between the size and the scope of a coalition: a coalition where
players cooperate on too many issues may be formed by a few players, which implies small spillovers
among them, whereas coalitions in which cooperation is restricted to few issues may be joined by many
players, thus raising many positive externalities within the coalition.
16
Bac and Raff (1996) focus on the effect of incomplete information about
bargaining strength on the choice of the bargaining procedure. The model involves two
players negotiating in a Rubinstein fashion over two pies, each of size one. The price-
surplus is known to agents and for both players the discount factor is assumed to be
equal over all issues. However, agents have asymmetric information about discounting
factors. One player is perfectly informed, while the other is uncertain about his
opponent's discounting factor. In particular, this can take one of the two values, H with
probability , and L with probability (1- ). This bargaining game has a sequential
equilibrium with rationalizing beliefs such that, while a weak (impatient) player prefers
to negotiate simultaneously over the two pies, a strong (patient) player may make an
offer on just one pie in order to signal bargaining strength. The uniformed player always
makes a combined offer on the two pies, which may include screening the informed
player and thus causing delay. According to this result, issue-by-issue negotiations may
thus arise from signaling considerations.
More recently, Fatima et al. (2003) studied the strategic behavior of agents by
using an agenda that is partly exogenous and partly endogenous. This is done by
decomposing the N issues into k equal stages. The issues for each stage are determined
exogenously, while the order in which issues are settled at each stage is determined
endogenously. The analysis shows that the negotiation outcome changes with the value
of k and that the optimal number of decompositions for an agent depends on the
negotiation parameters. In some negotiation scenarios the optimal value of k differs for
the two agents, while in others it is identical. In other words, there exist negotiation
scenarios where the utility to both agents can be improved by negotiating in stages
compared to the utilities they get from single-stage negotiations.
This result complements the explanations provided by the previous works,
namely that differing preferences over issues play an important role in determining
negotiation agendas. Exploring the agents' strategic behavior by separating negotiation
over the agenda from negotiation over the issues can be another promising line of
research.
17
4.3 Incomplete Information
Information can be defined as the knowledge about all factors that affect the ability of
an individual to make choices in any given situation. For example, in bargaining
between a buyer and a seller, information includes what an agent knows about its own
parameters (like his reservation price or his preferences over possible outcomes) and
what he knows about his opponent's parameters.
A critical assumption of the Rubinstein's (1982) alternating offer game is that
each player has complete information about the other's preferences. This assumption is
quite limiting because in real bargaining there are always some parameters agents are
uncertain about.
When incomplete information exists, new elements appear: a player, for
instance, may try to conclude from the other player's moves, who his opponents really
is; the other player, in turn, may try to bluff that he is tougher than he actually is, and so
on.
An important distinction in the ambit of incomplete information models is that
between symmetric and asymmetric information. Consider, for example, a game with
two players. The symmetric case corresponds to the situation in which both players lack
information about the opponent's parameters; in the asymmetric case, on the contrary,
uncertainty affects just one of the agents.
Following Harsanyi and Selten (1972), models of games with incomplete
information usually proceed by adopting the assumption that each player starts with the
same probability distribution on other players' private information and that these priors
are common knowledge. This is modeled by having the game begin with a probability
distribution, known to all players. Thus, agents not only have priors over other players'
private information, they also know what priors the other players have over their own
private information.
Starting from this idea, Rubinstein (1985) proposes an extension of his original
model to handle information uncertainty. This is a two person infinite horizon game that
considers incomplete information over agents' discounting factors. One of the players,
say player 2, may be one of two types: weak (for high discounting factor) and strong
(for low discounting factor). Player 1 adopts an initial belief about the identity of player
2. Player 1's preference is known to player 2. Agreement is reached in the first or
18
second time period. The main result of the work is the existence of a unique sequential
equilibrium when player 1's belief that player 2 is of type weak is higher than a certain
threshold, and another unique equilibrium when this belief is lower than the threshold.
Within a similar framework, Fundenberg and Tirole (1983, 1985) analyze a
buyer-seller infinite horizon bargaining game in which reservation prices are uncertain,
but time preferences are known. In particular, they focus on whether or not the
bargaining outcome can be ex-post efficient in the presence of one-sided and two-sided
uncertainty. When exactly one player's reservation value is her private information
(asymmetric case), the efficiency of the bargaining outcome depends on whether or not
the players' reservation values are independent of each other. If the players' reservation
values are independent, then the bargaining outcome can be ex-post efficient. If they
are, instead, correlated the bargaining outcome will not be efficient. When each player's
reservation value is her private information (symmetric case), the bargaining outcome
cannot be ex-post efficient whether or not the players' reservation values are
independent of each other.
Uncertainty over agent deadlines has been studied by Sandholm and Vulkan
(1999) in a symmetric information scenario. Since each player's deadline is private
information, there is a disadvantage in making offers. Any offer reveals some
information about the proposer's deadline, namely that it cannot be very long. If it were,
the proposer would stand a good chance of being able to out-wait the opponent, and
therefore would ask for a bigger portion of the surplus than it did. Similarly, the offerer
knows that it offered too much if the offer gets accepted: the offerer could have done
better by out-waiting the opponent. The main result of this work is that there exists a
sequential equilibrium where agents do not agree to a split until the first deadline, at
which time the agent with the later deadline receives the whole surplus. This result
holds both for pure and mixed strategies and, in most cases, is not affected by time
discounting and risk aversion.
In a more recent work, Fatima et al. (2002) address uncertainty over two
parameters: deadlines and reservation prices. In contrast with the previous models,
however, they assume that the probability distribution over these factors is private
knowledge for each player. As in Sandholm and Vulkan (1999), the optimal strategies
19
give the entire surplus of price to the agent with the longer deadline. However, time
discounting is not neutral anymore, but affects agents' payoffs.
To conclude, it is worth mentioning a model proposed by Petrakis and
Xepapadeas (1996) to study the problem of international environmental cooperation
under moral hazard. This work is more related to the literature on coalition formation,
but can provide interesting insights for the analysis and comprehension of bargaining
processes in the presence of information uncertainty. The set of players consists of the
following two groups of countries: environmentally conscious countries (ENCCs) and
less environmentally conscious countries (LENCCs). The authors analyze the
conditions under which the two groups can form a stable coalition to adjust emissions
so that a first-best global welfare optimum is achieved9.
The interesting aspect of the model is that asymmetries in information among
countries are considered, in the sense that countries entering into agreements know their
own emissions but cannot observe the emissions of the other participating countries.
This may create problems in the enforcement of the agreement, since countries have an
incentive to cheat by emitting more than the agreement stipulates. A mechanism that
detects cheating is developed by the authors in order to induce the desired emissions
even when the emission level of an individual country cannot be observed by the rest of
the participating countries.
From this brief review, it seems clear that the presence of uncertainty in
information may have a strong impact on the negotiation outcome and may provide an
appealing explanation for bargaining inefficiencies. Informational differences may also
explain the presence of bargaining power among agents and may have different effects
on the negotiation outcome when players are characterized by different degrees of risk
aversion.
4.4 Bargaining in stochastic environments
As discussed in the previous section, incomplete information refers to uncertainty over
players' parameters, such as players' discount factors, deadlines or reservation prices.
However, there are many other forms of uncertainty which may affect a bargaining
9In particular, a self-financing side payment scheme is determined, capable of securing a stable partial
coalition of ENCCs with a subset of LENCCs.
20
process. For example, the size of the `pie' over which agents are negotiating may vary
stochastically, as well as the disagreement point. These sources of uncertainty concern
the environment in which negotiations take place.
The theoretical literature on strategic bargaining in stochastic environment is
still quite limited, as well as its applications to existing data. In the last decade,
however, this issue has attracted increasing attention among researchers and various
efforts have been made in this direction.
In particular, Merlo and Wilson (1995) have proposed an extension of the basic
Rubinstein two-player alternating-offer game to a K-player bargaining model with
complete information, where both the identity of the proposer and the size of the pie
follow a stochastic process10.
In each period, a state is realized which determines the cake (i.e., the set of
possible utility vectors to be agreed upon in that period) and the order in which players
move. The selected player may either propose an allocation or pass. If he proposes an
allocation, each of the remaining players in turn accepts or rejects the proposal. If any
player rejects the proposal, a new state is realized and the process is repeated until some
proposed allocation is unanimously accepted.
More formally, the model can be described as follows. Let K= {1,..., K} denote
the set of players involved in the bargaining process and let S={s0,..., st} denote the set
of possible states of the world. A stochastic sequential bargaining game for K may be
indexed by (C, , ), where for each state s S, C(s) is a cake representing the set of
feasible utility vectors that may be agreed upon in that state, i(s) denotes the identity of
the player who makes the ith move in that state, and is the common discount factor for
the players.
The game is played as follows. Upon the realization of a state s, 1(s) (the player
who makes the first move in state s) chooses to either pass or propose an allocation in
C(s). If he proposes an allocation, player 2(s) responds by either accepting or rejecting
the proposal and after him, all the other players respond in the order prescribed by (s).
If the proposal is not unanimously accepted, but some players reject it, then the game
10More specifically, both random parameters follow a general Markov process, which is formally defined
as a discrete process in which the probabilities of transitions from one state to another are fixed and
independent of time that is, the system at time t+1 depends only on the system at time t, and not on the
state at any earlier time.
21
moves to the next period where a new state s' is realized according to a Markov process
=(0, 1, 2,...), defined in the space S. This procedure is then repeated except that the
order of moves is determined by (s') and the proposal must lie in the set C(s'). The
process continues until an allocation is proposed and accepted by all players.
An outcome of this bargaining game is either a pair (, ) where denotes the
period in which a proposal is accepted and denotes the proposed allocation which is
accepted in state s or disagreement. Then, for the game starting in state s, an outcome
(, ) implies a von Neumann-Morgenstern payoff to party i, E[i/ 0=s].
In order to solve the game, the authors focus on stationary sub-game perfect
outcomes and payoffs, that is, on outcomes and payoffs generated by a stationary sub-
game perfect strategy profile. The reason for this choice is that, when there are more
than two players, the game does not usually admit a unique equilibrium outcome but
multiple equilibria, even in the absence of uncertainty. As discussed in section 4.1,
stationary is the solution concept which is typically adopted in multilateral bargaining
models in order to solve the problem of indeterminacy of the negotiation outcome.
The main results of the paper can be summarized as follows:
R1: There exist a unique (stationary sub-game perfect) equilibrium;
R2: The equilibrium is efficient, even though it may involve delays.
This result is not exactly conforming to what the standard literature predicts. In
particular, according to the traditional models of bargaining, when an equilibrium
exists, either it is efficient and such that agreement is reached immediately (as in the
basic Rubinstein two-player game), or outcomes with delay may arise but efficiency is
not guaranteed anymore.
In the standard theory, the most common explanation for delaying agreement is
that players are unsure about the true preferences of their opponents. In other words,
incompleteness in information (see section 4.3) can cause inefficient equilibrium
outcomes. In the context of complete information, sequential bargaining models
generally admit delays only if there are multiple equilibria.
On the contrary, in the stochastic model by Merlo and Wilson, which is a model
with complete information, agreement may be delayed even in the unique stationary
sub-game perfect equilibrium and the equilibrium is still efficient.
22
The intuition for this result is that, when the future size of the cake is random,
there can be potential benefits to waiting as the size of the cake may grow in the future.
In other words, delay is caused by the expectation that the total bargaining value may
rise in the future and hence is efficient from the point of view of the negotiating parties.
Various applications exist of the framework described above, which mainly
focus on the problem of government formation. Merlo (1997), for instance, investigates
the process of government formation in post-war Italy, while Diermeier et al. (2004)
explore the role of bicameralism in determining government durability.
These studies seem to confirm the efficiency of delays predicted by Merlo and
Wilson for bargaining in stochastic environments. From a theoretical point of view,
however, this result depends also on other features of the game, such as the agreement
rule which is adopted, or the bargaining procedure. For example, when the agreement
rule is a general q-quota rule as in Eraslan and Merlo (2002) , uniqueness and efficiency
of the equilibrium are not guaranteed anymore. On the other hand, when players are
given the possibility to delay making offer, as in Furasawa and Wen (2001), the game
still has a unique equilibrium solution, but outcomes with delay are not efficient.
To conclude, it is important to notice that all the models mentioned above are
static models of bargaining, while the problem of negotiating over a pie of not-fixed
size could be better seen as a dynamic problem, where the state of the system evolves in
time. In Noncooperative bargaining theory, the dynamic nature of negotiations is
normally represented trough repeated bargaining games, which are described in the
following section. Further research is therefore needed in order to understand what are
the links between stochastic and dynamic nature of the bargaining setting, and how this
can be modeled.
4.5 Repeated Bargaining Situations
An implicit assumption of the Rubinstein's (1982) bargaining model is that players'
interaction ceases after a decision is reached, in other words, once the negotiation
process ends, players do not meet anymore. In fact, this is rarely the case in real settings
because agents usually have the opportunity to be involved in a sequence of bargaining
situations. Think, for example, of two adjacent countries and of the vast occasions of
23
bargaining they may have over time: from trade to international protection, from
political questions to environmental problems...
This section will focus on repeated bargaining games, which have been
proposed in the literature in the attempt to represent the long-term relationships that
may exist among bargainers. In a repeated framework, a game is played in successive
stages and at each stage players can decide on the basis of the actions and the outcomes
of the previous stages. There is an accumulation of information about the `history' of
the game that may affect players' strategic choices. In particular, even if it is the `same
game' which is repeated over a number of periods, the global `repeated game' becomes
a fully dynamic system with a much more complex structure than the one-stage game.
We will analyze here a simple repeated bargaining situation in which two
players sequentially bargain over the partition of an infinite number of cakes. The model
is based upon Muthoo (1995) and consists in an infinite repetition of the standard
Rubinstein model. Despite its simplicity, it provides some interesting results and allows
us to lay down the basic structure of repeated bargaining situations. The first important
qualification of the game is that players start bargaining over the partition of the (n+1)th
cake (where n=1, 2,...) if and only if they reach agreement on the partition of the nth.
The second qualification is that the time at which the players start bargaining over the
partition of the (n+1)th cake is determined by the time at which agreement is struck over
the partition of the nth cake. The structure of the game is as follows: there are two
agents A and B, who bargain over the partition of a cake of size ( >0), according to
an alternating-offer procedure. If agreement is reached at time t1, then immediately the
players consume their respective (agreed) shares. Then ( >0) time units later, at time
t2= t1+ , the players bargain over the partition of a second cake of size . Agreement at
time t2 is followed immediately with players consuming their agreed shares. This
process continues indefinitely (t3, t4,...), provided that players always reach agreement.
However, if players perpetually disagree over the partition of some cakes, then there is
no further bargaining over new cakes: agents have simply terminated their relationship.
In this model, the payoffs to the players depend on the number N of cakes that
they partition. In particular, if N=0 that is they perpetually disagree over the division
of the first cake then each player's payoff is zero. If N >0, then player i's payoff is:
24
x
N N
n tn n
i = xi exp(-ritn)
n=1 n=1
where xi is player i's share of the nth cake, tn is the time at which agreement
n
over the partition of the nth cake is struck, and i is player i's discount factor, with
i = exp(-ritn).
tn
This game has a unique stationary subgame perfect equilibrium and in
equilibrium agreement is reached immediately over the partition of each and every cake.
In general, however, this equilibrium outcome is different from the unique SPE partition
of the single available cake in Rubinstein's model. The intuition for this difference is as
follows: in a repeated bargaining model a player's discount factor determines not only
her cost of rejecting an offer, but also her value of future bargaining situations. Suppose
that player i becomes more patient (that is her discount rate ri decreases and her
discount factor i increases). This means that her cost of rejecting an offer decreases.
However, it also means that her value of future bargaining situations increases. When
bargaining over the partition of a cake, the former effect increases her bargaining power
(as she is more willing to reject offers), but the latter effect decreases her bargaining
power because she is more willing to accept offers so that the players can proceed to
bargain over the partition of the next cake. It has been shown that, under some plausible
conditions, the latter effect tends to dominate the former effect. This result implies that
when a player becomes less patient, she receives a greater share of each and every cake.
Thus, the impact of players' discount rates in repeated bargaining situations may differ
fundamentally from that in one-shot bargaining situations.
4.6 Synthesis of the results
Table 1 summarizes the results of the analysis conducted in sections 2 and 3. In
particular, three main characteristics of the equilibrium outcome are considered for the
basic Rubinstein alternating-offer game and its extensions: (i) the determinacy of the
equilibrium, (ii) the timing of the agreement and (iii) the efficiency of the result.
As previously noted, the model proposed by Rubinstein involves only two
players bargaining over the division of a single `pie' in a complete information setting.
Under these conditions, the alternating-offer bargaining game admits a unique SPE. In
25
such equilibrium, the agreement is reached immediately and the bargaining process is
efficient, in the sense that no resources are lost in delay.
Table 1: Characteristics of the equilibrium outcome in the basic Rubinstein model
and in the extensions analyzed through section 3.
Determinacy of the Timing of the Efficiency of the
equilibrium agreement equilibrium
Rubinstein (1982) Unique SPE No delay Guaranteed
Multiple players Multiple equilibria Possibility of delay Non-guaranteed.
Multiple issues
A) Mutually beneficial Unique equilibrium No delay Guaranteed
issues
B) Strictly controversial
issues:
B1) possibility of Unique equilibrium No delay Guaranteed
randomising
B2) no lotteries Multiple equilibria Possibility of delay Non-guaranteed
Incomplete information Unique equilibrium Presence of delay Not always guaranteed
Bargaining in a
stochastic environment
(uncertainty about the
size of the pie and/or the
order of moves)
· agreement rule
unanimity Unique equilibrium Possibility of delay The equilibrium is
efficient even with delay
· agreement rule Not normally guaranteed
different than Unique equilibrium Possibility of delay
unanimity
Repeated bargaining Unique equilibrium* No delay Not always guaranteed
games
*In general, however, this equilibrium outcome is different from the unique SPE of the basic Rubinstein's
model because of the different impact that players' discount rates usually have in repeated bargaining
situations. In particular, in the standard model, as a player becomes more patient, her share of the pie
increases, while the opposite happens in repeated bargaining games (see section 3.4.).
26
To which features of the model can we attribute this result? As shown in Table
1, one possible explanation for delaying agreement is that players are unsure about the
true preferences of their opponents. In other words, incompleteness in information can
cause inefficient equilibrium outcomes.
In a complete information setting, the presence of delay is closely related to the
existence of multiple equilibria, which may arise, for instance, when the negotiation
process involves more than two players. However, if the negotiation takes place in a
stochastic environment (such as, for example, when the size of the pie over which
players bargain varies stochastically) agreement may be delayed even in the unique sub-
game perfect equilibrium and the equilibrium may still be efficient11.
5 Noncooperative Coalition Theory
As noticed in Section 4.1, many real negotiations involve a large number of parties or
interest groups. When modelling these situations, the standard bargaining theory (both
cooperative and Noncooperative) makes the implicit assumption that there are just two
possible outcomes of the bargaining process: the cooperative outcome, where an
agreement is reached among all players involved in the negotiation process, and the
Noncooperative outcome, where no agreement forms. This dichotomy is often not
representative of real-life situations where partial agreements can form among a subset
of players.
In this section, we will focus on a different approach to multilateral negotiations,
which is `Noncooperative Coalition Theory' (NCT). Unlike standard bargaining theory,
this approach is able to take into account these intermediate cases because it allows for
the possibility of sub-coalitions to form.
More in general, we can distinguish both a cooperative and a Noncooperative
perspective within the theory of coalitions. However, as we will see in section 4.1,
cooperative coalition theory (CCT) basically coincides with the standard cooperative
11Another important element that may affect the timing of the solution is the presence of option values,
which do normally arise in dynamic contexts, that is when the state of the system evolves in time. This
aspect has been widely studied in optimal control theory, while research is still needed in the ambit of
non-cooperative game theory.
27
bargaining theory (Nash, 1950) for the case of N players and it cannot really help in
understanding the forces which drive the formation of (partial) coalitions.
As shown in some recent works by Gomes and Bloch, another reason to
concentrate on NCT is that this is, in general, more suitable to analyze the problem of
coalition formation within the context of negotiations, because it is more focused on
players' incentives to cooperate and on the procedures which lead to the formation of
coalitions.
5.1 Cooperative versus Noncooperative coalition theory
As emphasized by Bloch (1997), the analysis of endogenous formation of coalitions
poses three basic questions: (1) Which coalitions will form in equilibrium? (2) How will
the coalitional worth be divided among coalition members? (3) How does the presence
of other coalitions affect the incentives to cooperate?
The cooperative approach to coalition formation mostly focuses on the second
question, that is the division of the payoffs among co-operators, while the first question
is generally avoided. Most of cooperative models, indeed, are based on the idea that,
among all the possible coalitions that could form, the one that is most valuable12 will
actually be produced. Therefore, there is an assumption of Pareto-optimality (i.e., the
most efficient, value-maximizing coalition will always form) regardless of the process
required to form such a coalition. In fact, the processes are considered unproblematic, as
rational actors will always choose the outcome that maximizes their value. In other
words, once the best outcome is determined based on the attributes of the actors and the
payoffs available to them, the assumption is that the outcome will always realize.
This is exactly the same idea of Nash/cooperative bargaining theory which
imposes a number of axioms on the bargaining solution and, assuming that all players
participate in the agreement, focuses on the problem of dividing the pie according to
some criteria (such as feasibility, fairness, stability). The solution concepts adopted are
the same: from the Core to the Shapley-value, the Nucleolus, the Kalai-Smorodinsky
solution.
12Value is not usually defined explicitly, but is assumed to have some material weight. An example might
be the amount of policy power the winning coalition possesses.
28
The third question, dealing with competition between coalitions, is simply
ignored in traditional cooperative coalition theory (as well as in cooperative bargaining
theory, where competition among players is not really taken into account). The analysis
is based, indeed, on the characteristic function that assigns to each coalition C a real
number v(c) representing the worth of the coalition. The worth, however, is defined as
the aggregate payoff that a coalition can secure for itself irrespective of the behavior of
players outside the coalition. Then spillovers between coalitions are not allowed.
Because of these limitations, cooperative games, which were prevalent in earlier
coalition theory literature, have largely given way to Noncooperative games of coalition
formation. The Noncooperative approach is based on the partition function that assigns
an individual payoff to each player for each possible coalition structure. This is a
generalization of characteristic function games that allows for considerations of
spillovers. In particular, if the worth of a coalition C is independent of the coalitions
formed by the other players, the two definitions coincide. If, on the other hand, the
formation of coalitions affects all the players in the game, there is no univocal
relationship between partition functions and characteristic functions, and a game in
partition function form carries more information than a game in characteristic function
form.
In general, a Noncooperative game of coalition formation can be modeled as a
two-stage game: in the first stage, players decide non cooperatively whether or not to
join a coalition given the adopted burden-sharing rule; in the second stage, agents set
their policy/decision variables by maximizing their welfare function given the decision
taken in the first stage and the adopted burden-sharing rule. The standard assumption is
that coalition members act as a single player maximizing the aggregate payoff to their
coalition, but behave Noncooperatively towards outsiders. Equilibrium coalition
structures are then determined by applying the concept of internal and external stability
(Barrett 1994, 1997; Carraro and Siniscalco 1993; Hoel 1992; Hoel and Schneider 1997;
Rubio and Ulph, 2001). Internal stability means that no coalition member has an
incentive to leave its coalition to become a singleton, and external stability that no
29
singleton has an incentive to join a coalition, assuming that the remaining players do not
revise their membership decision13.
With this simple framework Noncooperative coalition theory can capture
players' incentive to cooperate without the need to make assumptions on the set of
possible outcomes, as standard bargaining theory does.
The two-stage approach described above represents the common denominator of
Noncooperative models of coalition formation. However, such models may differ
substantially with respect to other important features: the order of moves, the
membership rules, the players' conjectures, the type of free-riding in games with
spillovers, and so on. By changing these features of the game, the final coalition
structure changes.
5.2 Simultaneous (Noncooperative) games
A first important distinction is that between simultaneous and sequential
(Noncooperative) games of coalition formation. In simultaneous games, all players
announce at the same time their decision to form coalitions. In such games, it appears
that the set of Nash equilibria is often quite large, forcing researchers to use some
refinements in order to make interesting predictions. As noticed by Bloch (1997), these
refinements are usually of a cooperative nature; hence, the study of simultaneous games
of coalition formation is at the frontiers between cooperative and Noncooperative game
theory.
The problem of simultaneous formation of coalitions has been analyzed in the
literature under different coalition formation rules. Looking at the existing models, the
following three membership rules can be identified: (i) Open Membership, (ii) Exclusive
Membership, and (iii) Coalition Unanimity rules. A key difference between them lies in
what can happen to the membership of a coalition once it is formed: Can an existing
coalition break apart, admit new members or merge with other coalitions?
Open membership is the rule originally adopted in the literature on cartel
formation (D'Aspremont et al., 1983) and in the environmental literature on
13 Most of the existing contributions restrict coalition formation to a single coalition, allowing to group
players into signatories and non-signatories. However, there have been some recent developments that
admit the co-existence of multiple coalitions. These approaches invoke stability concepts that consider
not only deviations by single players but also by subgroups of players.
30
international agreements (Hoel, 1992; Carraro and Siniscalco, 1993; Barret 1994). In
open membership games, any player is free to join or leave a coalition. Accordingly,
players cannot specify in advance the coalition they wish to form. Rather, they
announce a message (for example, their willingness to participate in a coalition), and
coalitions are formed by all players who make the same announcement.
In exclusive membership games (Yi and Shin, 1994) or game (Hart and Kurz,
1983), each player can join a coalition only with the consensus of the existing members,
but she is free to leave the coalition. In this decision process, each player's message
consists in a list of players with whom she wants to form a coalition. Those who
announce the same list will then form a coalition, which is not, however, necessarily
formed by all players in the list.
Finally, in coalition unanimity games (Yi and Shin, 1994; Chander and Tulkens,
1997; Bloch, 1997) or game (Hart and Kurz, 1983), no coalition can form without the
unanimous consensus of its members. This implies that players are not free to either join
the coalition or to leave it. Therefore, this membership rule introduces restrictions both
on entry (as the exclusive membership rule) and on exit behaviors of players. In the
decision process, players' messages consist in a list of players as in the previous one.
However, if a coalition is formed, it is necessarily composed of all players in the list and
as soon as a player defects the coalition breaks up into singletons.
Yi (1997) provides an interesting analysis of the results of simultaneous games
of coalition formation for the different membership rules described above. In particular,
the author considers games where the formation of coalitions creates externalities on
non-members (which is often the case in real economic contexts) and recognizes in the
sign of the externalities a determinant organizing principle. In general, coalition
formation may create either positive or negative externalities on outside
members/coalitions. Examples of positive externalities include output cartels in
oligopoly and coalitions formed to provide public goods (such as environmental
quality). Examples of negative externalities are research coalitions with complementary
research assets and customs unions in international trade. The main results of the
analysis can be summarized as follows:
31
R1. With negative externalities, and under some reasonable conditions on the
partition function, the grand coalition is an equilibrium outcome under the Open
membership rule, but typically not under Exclusive Membership and Coalition
Unanimity.
R2. With positive externalities, the grand coalition is rarely an equilibrium
outcome for any of the membership rules mentioned above and only partial
agreements form. The grand coalition is more likely to emerge at the equilibrium
under Coalition Unanimity.
The explanation for these results is quite intuitive. If externalities are negative,
there is a disadvantage for players to stay outside the coalition and then it is more likely
that full cooperation is reached. On the contrary, if externalities are positive, players
who do not enter into the coalition may still enjoy (part of or all) the benefits from
cooperation without paying any cost. This produces incentives to free ride that, in turn,
prevent the formation of the grand coalition.
R3. In the presence of positive externalities, not only the grand coalition rarely
forms, but also the size of the partial agreement(s) which arise in equilibrium is
usually very small.
An important implication of these results is that standard bargaining theory may
not be appropriate in the presence of positive externalities where the emerging equilibria
are usually very far from full cooperation. Therefore, we can re-state that standard
bargaining theory can be appropriately applied to negotiations among n>2 players only
in the absence of externalities. With negative externalities, standard bargaining theory is
not appropriate, but results are equivalent to those obtained by using Noncooperative
game theory. With positive externalities, the only appropriate tool is Noncooperative
coalition theory.
The study of simultaneous games of coalition formation has, however, revealed
a number of difficulties which is important to underline. First of all, these games do not
usually admit a unique equilibrium outcome. The multiplicity of the equilibria imposes
the use of more refined solution concepts in order to obtain a sharp prediction about the
32
final coalition structure. Yi and Shin (1994) and Hart and Kurz (1983), for instance,
propose to consider cooperative refinements such as coalition-proof Nash equilibrium
and strong Nash equilibrium. These selection mechanisms, however, are in general very
stringent and this might generate unrealistic predictions on the final coalition structure.
Another limit of the simultaneous approach is that it does not allow identifying the
members of a coalition because all players have to decide at the same time whether or
not to participate. The identity of the players may instead be relevant for the
determination of the final equilibrium outcome. Finally, in simultaneous games, players
cannot be `farsighted' in the sense that individual deviations cannot be countered by
subsequent moves. Consider, for example, the departure of a player from a coalition. In
a simultaneous game, either the other coalition members remain together (in open
membership and in the game ) or the coalition breaks apart (in the game ). But in
both these formulations, members of the coalition which are left by the deviator are not
allowed to react to the move of the deviator.
5.3 Sequential (Noncooperative) games
The problems of simultaneous games have led to the formulation of sequential games of
coalition formation where the process is described by an explicit extensive form
Noncooperative game. In the context of games without spillovers, sequential processes
have been proposed by Selten (1981), Chatterjee et al. (1993), Moldovanu (1992) and
Perry and Reny (1994), among others. In most of these games, the basic structure is an
extension to n players of the Rubinstein's (1982) alternating-offer bargaining model
described in section 2. This structure was extended to games with spillovers by Bloch
(1996) and Ray and Vohra (1996).
All these works, although different with respect to the presence of externalities,
are based on a common assumption, which is: once a coalition has been formed, the
game is only played among the remaining players. The typical structure of the game is
as follows. Players are ordered according to a fixed rule14 and the first player starts by
proposing the formation of a coalition C to which she belongs. Each prospective
member responds to the proposal in the order determined by the fixed rule. If one of the
33
players rejects the proposal, she must make a counteroffer and propose a coalition C' to
which she belongs. If, instead, all proposed members accept, the coalition C is formed.
All players belonging to C then withdraw from the game, and the first player in N\C
starts making a proposal.
However, the assumption of immediate exit usually results in inefficient
outcomes, as shown in the following example inspired by Chatterjee et al. (1993). Let
n=3 and the gains from cooperation be represented by a coalitional function v(C)=0 if
C=1, v(C)=3 if C=2, and v(C)=4 when C=3. As players' discount factor, , converges
to 1, the outcome of the bargaining procedure where the grand coalition forms should
result in equal sharing of the coalitional surplus among the symmetric players (4/3 for
every player). But clearly, players then have an incentive to deviate forming an
inefficient coalition of size 2, which induces a payoff of 3/2 for each coalitional
member. If this coalition must leave the negotiation after its formation, the additional
surplus of 1 is lost.
In order to avoid these inefficiencies, other authors have proposed coalitional
bargaining models where agents cannot choose to exit, but they are given the possibility
to renegotiated over the formation of a coalition. In particular, Seidmann and Winter
(1998) have focused on games without externalities, while Gomes (2001) has extended
the analysis to the case of positive and negative spillovers. In these games with
continuous renegotiations, the grand coalition is ultimately formed, as players carry on
bargaining until all gains from cooperation are exhausted. However, delays may arise in
the enrichment of the agreement.
Unlike games with immediate exit, the models with continuous renegotiations do
usually produce efficient equilibrium outcomes.
5.4 Coalition Formation and Negotiations
Both the approaches described in the previous sections do not explicitly address an
important question, that is, when the members of a coalition would voluntarily choose
to leave the negotiation table. Many real-life situations seem to suggest that this
decision is a strategic action as much as the choice of forming coalitions. The Kyoto
14Okada (1996) proposes a model without externalities where players are randomly selected instead of
being ordered according to a fixed rule. Montero (1999) adopts a similar structure but allowing for the
34
protocol to reduce the emissions of greenhouse gases, for instance, shows that countries
often adopt this kind of strategies in the attempt to modify the final outcome of the
negotiation.
For the first time, these problems have been addressed in the literature in a work
by Bloch and Gomes (2003) where players are engaged in two parallel interactions: they
propose to form coalitions in order to extract gains from cooperation; and coalitions
participate in a repeated normal form game, where they choose endogenously when to
leave the negotiation process.
More precisely, the game, which is an infinite horizon N-player game, is
characterized by two distinct phases at every period. In the first phase, or contracting
phase, a player is chosen randomly to propose a coalition and a payment to all other
coalition members. Prospective members respond in turn to the offer and the coalition is
formed only if all its members agree to the contract. If a coalition is formed, the
proposer acquires control rights over the resources of coalition members (the proposer
player is then identified with the formed coalition). In the second phase, or action
phase, all proposer players choose an action, which may be a permanent action (in
which case the coalition they `control' exits the game) or a temporary action. The action
profile determines a flow payoff for all players, representing the underlying economic
opportunities. The interplay between the contracting and action phases enables the
authors to consider simultaneously issues of coalition formation, externalities and
endogenous exit decisions.
A key feature of the model is the existence of (pure) outside options for players
involved in the negotiation process. In classical two-player bargaining games, when an
agent chooses her outside option, negotiations end and the other player is left with a
fixed payoff. In multilateral negotiation contexts, when a player opts out and chooses to
enforce a permanent action, the other players continue to bargain over the formation of
coalitions and continue to choose actions which may affect the payoff of the exiting
player. The authors point out that there is a crucial distinction between situations where
outside option values are independent of the action of other players (pure outside
options) and situations where players' outside option values are affected by the actions
of remaining players.
existence of spillovers.
35
The main result of the paper is that there always exist an efficient equilibrium
outcome in games with pure outside options. The intuition for this result is as follows.
Early exit normally results in an aggregate efficiency loss. In a game with pure outside
options, players are able to capture this inefficiency loss and will never choose to leave
before the grand coalition is formed. By staying in the game one more period, indeed, a
player is guaranteed to obtain her outside option (which remain available because
outside options are pure), and is able to capture the inefficiency loss by proposing to
form the grand coalition when she is recognized to make an offer. Hence, early exit will
never occur in equilibrium.
The authors also provide some examples of games where the outside options are
not pure. They show that, in such cases, the equilibrium outcomes may lead to the
inefficient formation of partial coalitions. This result highlights the difference and the
importance of this model with respect to the coalitional bargaining models previously
mentioned. In a setting with externalities, for instance, Ray and Vohra (1999) showed
that when players cannot renegotiate, the outcome of coalition formation is typically
inefficient, as players have an incentive to leave the game before extracting the entire
surplus. On the contrary, Gomes (2001) established that when renegotiation occurs and
players cannot choose to exit, the outcome is always efficient. Bloch and Gomes (2003)
identify a new type of friction externalities on players' endogenous outside options
that may lead to bargaining inefficiencies.
6 Fair-division theory
Starting from the basic Rubinstein's alternating-offer game, almost all economic models
of bargaining have remained faithful to the traditional assumptions about agent behavior
underlying the economic science, that are: perfect rationality and purely selfish pursuit
of personal interests. In other words, standard bargaining theory assumes that when
deciding whether or not to accept an offer, each bargainer focuses exclusively on her
own payoff and compares what she can get by accepting the proposal with what she
could get by rejecting it and moving to next period. According to this framework,
agents do not have any fairness concern, in the sense that they do not care about the
distribution of payoffs or the intentions of the other bargainers.
36
Yet everyday experience indicates that fairness consideration may have a
significant influence on people's behavior, and that humans are inclined to retaliate
against those who treat them unfairly.
6.1 Experimental Evidence
Traditional assumptions of perfectly rational and self-oriented agents do normally work
very well in the context of `almost' perfectly competitive markets, where the number of
players is `very big' and what really matters for the economic science is the
representative agent, i.e. an imaginary agent whose every single trait of character is the
average of that trait over all agents present in the market (see, for instance, the
experimental work by Roth et al, 1991). On the contrary, several experimental studies
of bargaining situations have revealed the importance of fairness considerations in
negotiation contexts. Negotiations are indeed a very peculiar type of economic
interactions, because in their case the assumptions of perfect competition and `large
numbers' are inappropriate.
Most of the existing bargaining experiments examine one-period (or "ultimatum")
games. In such games, the Proposer makes a take-it-or-leave-it offer to the respondent
on how they should split a surplus of a fixed size. The bargaining proceeds as follows:
the proposer offers a share s to the respondent (with a share 1-s going to herself); the
offer can be accepted in which case the respondent gets a payoff of x2=s, and the
proposer gets a payoff of x1=1-s; or it can be rejected, in which case both players get a
payoff of 0. The standard model predicts that the unique sub-game perfect equilibrium
for this game is for the proposer to offer s=0, which is accepted by the respondent. This
outcome is Pareto efficient, but it is clearly highly unequal.
The data generated by ultimatum experiments with complete information indicate
that rather than making offers where the proposers keep the entire surplus minus the
smallest unit of account, the proposers offer distributions that are closer to an equal split
of the surplus (see, for instance, the experimental results of Thaler (1988); Güth and
Tietz (1990); Roth (1995), Slonim and Roth (1997), and Ochs and Roth (1989); Spegel
et al. (1990)).
Regularities and robust facts emerging from experimental studies of this type are:
(i) there are virtually no offers above 0.5; (ii) most of the offers falls within the 0.4-0.5
37
interval; (iii) there are almost no offers below 0.2; (iv) low offers are usually rejected,
with the probability of rejection being inversely related to s.
Similar experimental results are obtained for "dictator" games and public good
distribution games. In the former, a dictator has to decide what share s of a given
surplus he should give to his opponent: whereas the standard model predicts that s=0,
experimental evidence indicate that around half of the subjects choose 0~~2 players needs to be such that there exists a
strategy for each player, which guarantees him or her a piece of the surplus that s/he
considers equal to the largest, no matter what the other players do. None of the n-person
proportional procedures is envy-free: whilst they guarantee each player a portion which
is at least 1/n, one or more of the players may think that another player received a larger
piece.
In addition to not being envy-free in the case of n>2 players, proportional fair
allocation procedures cannot easily be employed in the case of non-divisible goods. In
this case, the main allocation procedures is the Knaster's sealed bids procedure, an n-
person auction scheme that is proportional and efficient, but not envy-free for n>2.
Players submit sealed bids for items, which are then allocated to the highest offerer; in
the second stage, there are some side-payments, with the monetary reallocation
estimated by computing the "fair share" for each player.
6.3.2 Refinements of the basic procedures
Generally, there is no fair-division scheme that is simultaneously (1) algorithmic;
(2) proportional; and (3) efficient. An exception is the Adjusted Winner procedure
(AW) (Brams and Taylor 1996, 2000), which produces settlements that are efficient,
envy-free and equitable with respect to bargainers' announced preferences for n=221.
However, the AW provides no incentives for player to be truthful about their
preferences: it is once again the information structure which determines the properties
of the solution. In this procedure, two parties begin by independently distributing a total
of 100 points across all items to be allocated, according to their own valuation of the
goods. Each player is then assigned the goods which s/he values most. The initial
allocation is then adjusted to equalize the total valuations of the goods for the two
42
players22. The allocation thus achieved is efficient no player can be made better off
without the other being worse off; it is equitable, in that announced valuations are
equated; and it is envy-free no player would trade his or her allocation for that of the
other player. However, envy-freeness and equitability are only apparent, as they rely on
the truthful revelation of players' valuations with asymmetric information, the player
with complete information can exploit the other player and manipulate the procedure.
An alternative envy-free allocation which is however not efficient is the
Proportional Allocation (PA). As the name indicates, under this procedure players are
allocated the same share of their valuation of the goods, hence the resulting allocation is
envy-free. Under PA, players have the incentive to reveal near true preferences, as the
payoffs are hardly affected by deviations23.
PA's incentives to be truthful come at an efficiency cost with respect to AW: it is
however possible to induce players to reveal their nearly-true valuation under an AW
procedure, by imposing a PA allocation as a default, should either player be dissatisfied
with the allocation reached under AW.
An alternative is Raith's Adjusted Knaster (AK) procedure (Raith, 2000), which is
a combination of the Knaster and the AW procedures. AK combines the efficient side
payments of the sealed bid procedure with the equitability conditions of AW, for 2
players. By imposing an equitable monetary transfer, the AK implements an outcome
that is at least as good as that of the AW.
The fact that these procedures ensure efficiency, equitability and envy-freeness in
the 2 persons case is encouraging, despite AW's theoretical (but probably not practical
24) vulnerability to strategic manipulation. Unfortunately, neither AW nor PA maintain
these properties when there are n>2 players. Algorithms have been developed that find
an allocation satisfying two of the three properties: which pair of properties constitutes
the most desirable set is not clear a priori.
21The AW, in its basic form, implements the KalaiSmorodinsky solution (Raith 2000).
22The same argument applies to issues negotiated (continuous vs. discrete), where player 1 gets 60% of
the issue means that the issue is resolved 60% in favour of player 1.
23In fact, it is shown in Brams and Taylor (1996), p. 77, that, in the absence of reliable information about
the opponent's preferences, it is a dominant strategy for each player to reveal a valuation close to his or
her true valuation especially in the range 20-80. For extreme values, truth revelation is a dominant
strategy if players have symmetric or opposite valuation of the good.
24See Brams and Taylor (1996), p. 85.
43
When the number of players increases, the algorithms for envy-free allocation get
very complicated. There are modifications of other procedures which generate near
envy-free allocations, with the degree of error being within any present tolerance level
(Brams and Taylor, 1996, p.129). For instance, the general moving knife procedure can
be modified to allow players to re-enter the game even though they have received a
piece of the cake, by calling cut again and again, with the provision that they must take
the piece of cake determined by their most recent cut, and return the previous piece.
In a generalization of the divide-and-choose procedure, players can achieve an
envy-free allocation of part of the cake25 through a trimming strategy: at different stages
of the game, parties create equal shares for themselves by trimming others' partitions of
the surplus. Note that the final allocation will depend on the order in which players
move there are therefore many envy-free allocations, and many possible equilibria to
this game. Moreover, not only does this procedure become very complicated as n
increase, but also it is not clear what to do with the trimming and the piece left aside:
these cannot be distributed, as in the case for n=3, in a manner that leaves players envy-
free, and exhausts the cake. If one allows for an infinite number of stages, then the
procedure can be applied over and over again and eventually the whole cake is
allocated. But the corresponding finite algorithm is complex and unbounded in the sense
that the number of cuts needed to produce a given division depends on the number of
players and on their preferences. Within this procedure, the existence of an envy-free
allocation rests on the assumption that the good is divisible or, in the case of
indivisible goods, that there are enough of more divisible goods which can be trimmed
in lieu of the discrete good.
Brams and Kilgour (2001) propose a fair division procedure for the allocation of
indivisible goods and divisible bads (the price to be paid for the goods). In the Gap
procedure, goods are assigned to players in such a way that the total sum of their bids
for the goods is maximized (maxsum allocation); the prices paid are obtained by
decreasing bid values to the next highest bids until their sum is equal to or less than the
total value of the goods: once this level is reached (and provided that the sum is not
equal to it) reductions in the next higher bids are made in proportion to the differences
25This is an extension of the Selfridge-Conway discrete procedure, which allows an envy-free allocation
of a heterogeneous good among three players with different valuation of the good in question.
44
between these bids for each good and the next lower bids. Bids therefore serve the dual
purpose of assigning the goods to the players, and determining the prices players have
to pay: however, unlike in the Knaster's procedure, the highest bidders do not
necessarily receive the goods, and the prices players pay depend not only on their own
evaluation of the good, but also on other players' the more competitive the bids, the
higher the price to pay will be. Under the Gap procedure, no players ever pays a
negative price the lowest price a player can pay being the lowest bid for that good; no
player pays more than his bid players pay either their own bid, or a lower price; the
allocation is Pareto efficient, because it maximizes total surplus.
However, the Gap procedure does not produce envy-free allocations that is, a
player may prefer the good assigned to another player, at the price that the other player
pays for it, to the one assigned to him, at the Gap price. Potthoff (2002) proposes a
linear programming solution to find an envy free solution which is closes to the Gap
solution that is, that set of envy-free prices that minimizes the sum of absolute
difference from the Gap prices. Such a solution always exists when negative prices are
allowed but its existence is not guaranteed otherwise
6.4. Synthesis of the procedures
Whatever the underlying motivations are for the emergence of "fair" behaviors and/or
outcomes, the perception of fairness is critical to facilitate the achievement of an
agreement on how to divide a surplus in a Noncooperative negotiation framework,
where allocations need to be self-enforcing.
In
Table 2, the main fair division procedures presented here are summarized and
compared, with respect to three main characteristics: equity, envy-freeness, and
efficiency. An allocation is equitable when players think that their portion is worth the
same as everybody else's'; it is envy-free when every player thinks s/he receives a
portion that is at least tied for the largest, or tied for most valuable, and hence does not
envy any other player; and it is efficient, if no player can be made better off, without
another player being made worse off.
45
When the properties of the fair division procedures vary depending on the
number of players (two or more than two) and/or the type of item they can be applied to
(homogenous or heterogeneous, divisible or indivisible), this is emphasized in the table.
Table 2: Summary of the main fair division procedures, and key characteristics
Players Surplus Equity Envy-free Efficient
Basic Fair Division Procedures
Discrete procedures
Basic divide and 2 Divisible Proportional Yes if no No unless
choose Heterogeneous asymmetries in players have
information symmetric
preferences for
all the parts of
the cake
Filer and choose 2 Non-divisible Proportional Yes if no No unless
public good asymmetries in players have
information symmetric
preferences for
all the parts of
the cake
Discrete trimming >2 Non-divisible Proportional Yes No
Selfridge-Conway 3 Divisible, Proportional Yes No
discrete heterogeneous
Lone-divider >2 Divisible Proportional No No unless
Heterogeneous players have
symmetric
preferences for
all the parts of
the cake
Lone-chooser 2 and Divisible Proportional Yes for 2 No unless
>2 players. players have
symmetric
Otherwise no. preferences for
all the parts of
the cake
Continuous procedures
Moving knife 2 Divisible Proportional Yes if no No unless
Heterogeneous asymmetries in players have
information symmetric
preferences for
all the parts of
the cake
Generalized >2 Divisible Proportional No No unless
moving knife players have
symmetric
preferences for
all the parts of
the cake
46
Last diminisher >2 Divisible Proportional No No unless
players have
symmetric
preferences for
all the parts of
the cake
Refinements of the basic procedures
Adjusted winner 2 Divisible Proportional Yes with Yes
Indivisible respect to
players stated
preferences
>2 Divisible It can satisfy two of the properties only
Indivisible
Proportional 2 and Divisible Proportional Envy-free No
allocation >2
Adjusted >2 Divisible Proportional Envy-free No
Knaster's Indivisible
procedure
Gap procedure >2 Indivisible goods Proportional Envy-free Yes
(max-sum and divisible bads
allocation)
The procedures described in this short review are applicable to both homogenous
and heterogeneous players it is in fact the structure of information which determines
the properties of the solution. When (a)symmetry of information and players' preference
structure affect the properties of the solution, this is highlighted in the table.
There are therefore numerous fair-division procedures, which exhibit different
properties with respect to the efficiency, equitability, envy-freeness of both the
procedures and the resulting allocation. It is difficult to answer theoretically which
procedure is best, as trade-offs among their characteristics, as well as consideration of
vulnerability of the procedure to strategic manipulation, need to be considered.
However, the focus of researcher and practitioners should shift away from the
achievement of an efficient allocation as the overriding goal, and pay more attention to
the properties of equity and envy-freeness which should be satisfied, if a self-
enforcing agreement is needed. In fact, restricting the possible agreements to those
satisfying some form of equity and envy-freeness could help select one equilibrium
when a multiplicity of equilibria could be possible.
47
7 Conclusions
The relevance of negotiations to everyday life cannot be overemphasized. Yet, a
comprehensive theory of negotiation is still missing: the factors involved in the
processes of negotiations are so complex and varied, that they have been tackled in
isolation, with the consequence that many theoretical results of the standard models do
not always find support in empirical evidence.
From this review of the theory four main considerations emerge, which should
be taken into account in the formulation of a suitable negotiation model:
The Noncooperative approach to negotiations is useful in that it allows for the
analysis of players' incentives to cooperate. Moreover, the outcome of a
Noncooperative game has the property of being self-enforcing. This is particularly
important at the international level where there are no supranational governing
bodies that can impose cooperation, and agreements have to be reached voluntarily
among sovereign states.
The sequential-move approach enables the process of negotiation to be modeled.
This, in turn, allows for the analysis of some particular issues (such as bargaining
and political power, asymmetric information, time preferences) which may have
relevant effects on the bargaining outcome.
However, standard bargaining theory is not well suited to deal with bargaining
situations where (positive) externalities are involved. The presence of externalities
opens up the possibility of intermediate agreements, neither fully cooperative, nor
fully Noncooperative. These more complex situations can be better explored by
Noncooperative coalition theory.
Finally, both standard bargaining theory and coalition theory do not address the
issue of fair division in a comprehensive manner, focusing almost exclusively on the
efficiency property of the outcomes. The integration of fair division theory in
negotiation is however crucial if the solution/agreement is to be implemented and
sustained.
48
8 References
1. Alesina, A., I. Angeloni and F. Etro (2001), `The political economy of unions', NBER
Working Paper, December 2001.
2. Andreoni, J. and J. H. Miller (1996), `Giving according to GARP: an experimental study of
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