Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division of payoffs to choose. Such surplus-sharing problems (also called bargaining problem) are faced by management and labor in the division of a firm's profit, by trade partners in the specification of the terms of trade, and more.
The present article focuses on the ''normative'' approach to bargaining. It studies how the surplus ''should'' be shared, by formulating appealing axioms that the solution to a bargaining problem should satisfy. It is useful when both parties are willing to cooperate in implementing the fair solution. The five axioms, any Nash Bargaining Solution should satisfy are Pareto Optimality (PAR), Individual Rationality (IR), Independent of Expected Utility Representations (INV), Independence of Irrelevant Alternatives (IIA) and Symmetry (SYM). While SYM and PAR restrict the behavior of the solution to only a specific bargaining problem, INV and IIA require consistency of solution across bargaining problems in Game Theory. Such solutions, particularly the Nash solution, were used to solve concrete economic problems, such as management–labor conflicts, on numerous occasions.
An alternative approach to bargaining is the ''positive'' approach. It studies how the surplus is actually shared. Under the positive approach, the bargaining procedure is modeled as a non-cooperative game. The most common form of such game is called ''
sequential bargaining
Sequential bargaining (also known as alternate-moves bargaining, alternating-offers protocol, etc.) is a structured form of bargaining between two participants, in which the participants take turns in making offers. Initially, person #1 has the rig ...
''.
Formal description
A two-person bargain problem consists of:
* A feasibility set
, a closed subset of
that is often assumed to be convex, the elements of which are interpreted as agreements.
* A disagreement, or threat, point
, where
and
are the respective payoffs to player 1 and player 2, which they are guaranteed to receive if they cannot come to a mutual agreement.
The problem is nontrivial if agreements in
are better for both parties than the disagreement point. A solution to the bargaining problem selects an agreement
in
.
Feasibility set
The feasible agreements typically include all possible joint actions, leading to a feasibility set that includes all possible payoffs. Often, the feasible set is restricted to include only payoffs that have a possibility of being better than the disagreement point for both agents.
Disagreement point
The disagreement point
is the value the players can expect to receive if negotiations break down. This could be some
focal equilibrium that both players could expect to play. This point directly affects the bargaining solution, however, so it stands to reason that each player should attempt to choose his disagreement point in order to maximize his bargaining position. Towards this objective, it is often advantageous to increase one's own disagreement payoff while harming the opponent's disagreement payoff (hence the interpretation of the disagreement as a threat). If threats are viewed as actions, then one can construct a separate game wherein each player chooses a threat and receives a payoff according to the outcome of bargaining. It is known as ''Nash's variable threat game''.
Nash bargaining game
John Forbes Nash
John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow g ...
was the first to study cooperative bargaining. His solution is called the
Nash bargaining solution Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division o ...
. It is the unique solution to a two-person bargaining problem that satisfies the axioms of ''scale invariance'', ''symmetry'', ''efficiency'', and ''independence of irrelevant alternatives''. According to Walker, Nash's bargaining solution was shown by
John Harsanyi
John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994.
He is best known for his contributions to the ...
to be the same as
Zeuthen
Zeuthen is a municipality in the district of Dahme-Spreewald in Brandenburg in Germany.
Geography
It is located near the southeastern Berlin city limits on the western shore of the Dahme River and the Zeuthener See. It borders Eichwalde in the ...
's solution of the bargaining problem.
The Nash bargaining game is a simple two-player game used to model bargaining interactions. In the Nash bargaining game, two players demand a portion of some good (usually some amount of money). If the total amount requested by the players is less than that available, both players get their request. If their total request is greater than that available, neither player gets their request.
Nash (1953) presents a non-cooperative demand game with two players who are uncertain about which payoff pairs are feasible. In the limit as the uncertainty vanishes, equilibrium payoffs converge to those predicted by the Nash bargaining solution.
Equilibrium analysis
Strategies are represented in the Nash demand game by a pair (''x'', ''y''). ''x'' and ''y'' are selected from the
interval 'd'', ''z'' where ''d'' is the disagreement outcome and ''z'' is the total amount of good. If ''x'' + ''y'' is equal to or less than ''z'', the first player receives ''x'' and the second ''y''. Otherwise both get ''d''; often
.
There are many
Nash equilibria
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
in the Nash demand game. Any ''x'' and ''y'' such that ''x'' + ''y'' = ''z'' is a Nash equilibrium. If either player increases their demand, both players receive nothing. If either reduces their demand they will receive less than if they had demanded ''x'' or ''y''. There is also a Nash equilibrium where both players demand the entire good. Here both players receive nothing, but neither player can increase their return by unilaterally changing their strategy.
In Rubinstein's alternating offers bargaining game,
players take turns acting as the proposer for splitting some surplus. The division of the surplus in the unique subgame perfect equilibrium depends upon how strongly players prefer current over future payoffs. In particular, let d be the discount factor, which refers to the rate at which players discount future earnings. That is, after each step the surplus is worth d times what it was worth previously. Rubinstein showed that if the surplus is normalized to 1, the payoff for player 1 in equilibrium is 1/(1+d), while the payoff for player 2 is d/(1+d). In the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.
Bargaining solutions
Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.
Nash bargaining solution
John Forbes Nash Jr.
John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow game ...
proposed that a solution should satisfy certain axioms:
#Invariant to affine transformations or Invariant to equivalent utility representations
#
Pareto optimality
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engine ...
#
Independence of irrelevant alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
#
Symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
Nash proved that the solutions satisfying these axioms are exactly the points
in
which maximize the following expression:
::
where ''u'' and ''v'' are the utility functions of Player 1 and Player 2, respectively, and d is a disagreement outcome. That is, players act as if they seek to maximize
, where
and
, are the
status quo
is a Latin phrase meaning the existing state of affairs, particularly with regard to social, political, religious or military issues. In the sociological sense, the ''status quo'' refers to the current state of social structure and/or values. W ...
utilities (the utility obtained if one decides not to bargain with the other player). The product of the two excess utilities is generally referred to as the ''Nash product''. Intuitively, the solution consists of each player getting their status quo payoff (i.e., noncooperative payoff) in addition to a share of the benefits occurring from cooperation.
Kalai–Smorodinsky bargaining solution
Independence of Irrelevant Alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
can be substituted with a
Resource monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM pri ...
axiom. This was demonstrated by
Ehud Kalai
Ehud Kalai is a prominent Israeli American game theorist and mathematical economist known for his contributions to the field of game theory and its interface with economics, social choice, computer science and operations research. He was the J ...
and Meir Smorodinsky. This leads to the so-called ''Kalai–Smorodinsky bargaining solution'': it is the point which maintains the ratios of maximal gains. In other words, if we normalize the disagreement point to (0,0) and player 1 can receive a maximum of
with player 2's help (and vice versa for
), then the Kalai–Smorodinsky bargaining solution would yield the point
on the Pareto frontier such that
.
Egalitarian bargaining solution
The egalitarian bargaining solution, introduced by Ehud Kalai, is a third solution which drops the condition of scale invariance while including both the axiom of
Independence of irrelevant alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
, and the axiom of
resource monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM pri ...
. It is the solution which attempts to grant equal gain to both parties. In other words, it is the point which maximizes the minimum payoff among players. Kalai notes that this solution is closely related to the
egalitarian
Egalitarianism (), or equalitarianism, is a school of thought within political philosophy that builds from the concept of social equality, prioritizing it for all people. Egalitarian doctrines are generally characterized by the idea that all hum ...
ideas of
John Rawls
John Bordley Rawls (; February 21, 1921 – November 24, 2002) was an American moral, legal and political philosopher in the liberal tradition. Rawls received both the Schock Prize for Logic and Philosophy and the National Humanities Medal in 1 ...
.
Comparison table
Experimental solutions
A series of experimental studies found no consistent support for any of the bargaining models. Although some participants reached results similar to those of the models, others did not, focusing instead on conceptually easy solutions beneficial to both parties. The Nash equilibrium was the most common agreement (mode), but the average (mean) agreement was closer to a point based on expected utility. In real-world negotiations, participants often first search for a general bargaining formula, and then only work out the details of such an arrangement, thus precluding the disagreement point and instead moving the focal point to the worst possible agreement.
Applications
Kenneth Binmore
Kenneth George "Ken" Binmore, (born 27 September 1940) is an English mathematician, economist, and game theorist, a Professor Emeritus of Economics at University College London (UCL) and a Visiting Emeritus Professor of Economics at the Unive ...
has used the Nash bargaining game to explain the emergence of human attitudes toward
distributive justice
Distributive justice concerns the socially just allocation of resources. Often contrasted with just process, which is concerned with the administration of law, distributive justice concentrates on outcomes. This subject has been given considera ...
. He primarily uses
evolutionary game theory
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Ma ...
to explain how individuals come to believe that proposing a 50–50 split is the only
just
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
solution to the Nash bargaining game.
Herbert Gintis
Herbert Gintis (February 11, 1940 – January 5, 2023) was an American economist, behavioral scientist, and educator known for his theoretical contributions to sociobiology, especially altruism, cooperation, epistemic game theory, gene-culture co ...
supports a similar theory, holding that humans have evolved to a predisposition for
strong reciprocity Strong reciprocity is an area of research in behavioral economics, evolutionary psychology, and evolutionary anthropology on the predisposition to cooperate even when there is no apparent benefit in doing so. This topic is particularly interesting ...
but do not necessarily make decisions based on direct consideration of utility.
Bargaining solutions and risk-aversion
Some economists have studied the effects of
risk aversion
In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
on the bargaining solution. Compare two similar bargaining problems A and B, where the feasible space and the utility of player 1 remain fixed, but the utility of player 2 is different: player 2 is more risk-averse in A than in B. Then, the payoff of player 2 in the Nash bargaining solution is smaller in A than in B.
However, this is true only if the outcome itself is certain; if the outcome is risky, then a risk-averse player may get a better deal as proved by
Alvin E. Roth
Alvin Eliot Roth (born December 18, 1951) is an American academic. He is the Craig and Susan McCaw professor of economics at Stanford University and the Gund professor of economics and business administration emeritus at Harvard University. and
Uriel Rothblum
Uriel George "Uri" Rothblum (Tel Aviv, March 16, 1947 – Haifa, March 26, 2012) was an Israeli mathematician and operations researcher. From 1984 until 2012 he held the Alexander Goldberg Chair in Management Science at the Technion – Israel Ins ...
Further reading
For a comprehensive discussion of the
Nash bargaining solution Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division o ...
and the huge literature on the theory and application of bargaining - including a discussion of the classic
Rubinstein bargaining model
A Rubinstein bargaining model refers to a class of bargaining games that feature alternating offers through an infinite time horizon. The original proof is due to Ariel Rubinstein in a 1982 paper. For a long time, the solution to this type of game ...
- see
Abhinay Muthoo
Abhinay Muthoo is an economist specializing in negotiations, game theory and public policy.
Abhinay has 37 years of teaching and research experience, across several universities including Bristol, Cambridge, Essex, Warwick and the LSE. He has b ...
's book Bargaining Theory and Application.
Abhinay Muthoo
Abhinay Muthoo is an economist specializing in negotiations, game theory and public policy.
Abhinay has 37 years of teaching and research experience, across several universities including Bristol, Cambridge, Essex, Warwick and the LSE. He has b ...
Bargaining Theory with Applications
Cambridge University Press
1999.
See also
*
Bargaining
In the social sciences, bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service debate the price or nature of a transaction. If the bargaining produces agreement on terms, the transaction takes plac ...
*
Rubinstein bargaining model
A Rubinstein bargaining model refers to a class of bargaining games that feature alternating offers through an infinite time horizon. The original proof is due to Ariel Rubinstein in a 1982 paper. For a long time, the solution to this type of game ...
*
Sequential bargaining
Sequential bargaining (also known as alternate-moves bargaining, alternating-offers protocol, etc.) is a structured form of bargaining between two participants, in which the participants take turns in making offers. Initially, person #1 has the rig ...
*
Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
*
Ultimatum game
The ultimatum game is a game that has become a popular instrument of economic experiments. An early description is by Nobel laureate John Harsanyi in 1961. One player, the proposer, is endowed with a sum of money. The proposer is tasked with s ...
References
*
External links
Nash bargaining solutions
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Game theory game classes
Cooperative games
Bargaining theory