The two-person **bargaining problem** studies how two agents share a surplus that they can jointly generate. It is in essence a payoff selection problem. 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. There are two typical approaches to the bargaining problem. The normative approach studies how the surplus should be shared. It formulates appealing axioms that the solution to a bargaining problem should satisfy. The positive approach answers the question how the surplus will be shared. Under the positive approach, the bargaining procedure is modeled in detail as a non-cooperative game.

- The bargaining game
- Formal description
- Feasibility set
- Disagreement point
- Equilibrium analysis
- Bargaining solutions
- Nash bargaining solution
- Kalai–Smorodinsky bargaining solution
- Egalitarian bargaining solution
- Comparison table
- Experimental solutions
- Applications
- Bargaining solutions and risk-aversion
- See also
- References
- External links

The ** Nash bargaining solution** 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,^{ [1] } Nash's bargaining solution was shown by John Harsanyi to be the same as Zeuthen's solution^{ [2] } 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.^{ [3] }

Rubinstein also modelled bargaining as a non-cooperative game in which two players negotiate on the division of a surplus known as the alternating offers bargaining game.^{ [4] } The players take turns acting as the proposer. The division of the surplus in the unique subgame perfect equilibrium depends upon how strongly players prefer current over future payoffs. In the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.

For a comprehensive discussion of the Nash bargaining solution and the huge literature on the theory and application of bargaining - including a discussion of the classic Rubinstein bargaining model - see Abhinay Muthoo's book Bargaining Theory and Application.^{ [5] }

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. is often assumed to be convex because, for any two feasible outcomes, a convex combination (a weighted average) of them is typically also feasible.
- 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 .

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 the agents that are bargaining.^{ [3] }

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.

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 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,^{ [4] } 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.

Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.

John Nash proposed^{ [6] } that a solution should satisfy certain axioms:

- Invariant to affine transformations or Invariant to equivalent utility representations
- Pareto optimality
- Independence of irrelevant alternatives
- Symmetry

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 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.^{ [7] }^{:15–16}

Independence of Irrelevant Alternatives can be substituted with a Resource monotonicity axiom. This was demonstrated by Ehud Kalai and Meir Smorodinsky.^{ [8] } 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 .

The egalitarian bargaining solution, introduced by Ehud Kalai,^{ [9] } is a third solution which drops the condition of scale invariance while including both the axiom of Independence of irrelevant alternatives, and the axiom of resource monotonicity. 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 ideas of John Rawls.

Name | Pareto-optimality | Symmetry | Scale-invariance | Irrelevant-independence | Resource-monotonicity | Principle |
---|---|---|---|---|---|---|

Nash (1950) | Maximizing the product of surplus utilities | |||||

Kalai-Smorodinsky (1975) | Equalizing the ratios of maximal gains | |||||

Kalai (1977) | Maximizing the minimum of surplus utilities |

A series of experimental studies^{ [10] } 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.^{ [11] } 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.

Kenneth Binmore has used the Nash bargaining game to explain the emergence of human attitudes toward distributive justice.^{ [12] }^{ [13] } He primarily uses evolutionary game theory to explain how individuals come to believe that proposing a 50–50 split is the only just solution to the Nash bargaining game. Herbert Gintis supports a similar theory, holding that humans have evolved to a predisposition for strong reciprocity but do not necessarily make decisions based on direct consideration of utility.^{ [14] }

Some economists have studied the effects of risk aversion 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.^{ [15] }^{:303–304} 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 and Uriel Rothblum ^{ [16] }

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

**Bargaining** or **haggling** is a type of negotiation in which the buyer and seller of a good or service debate the price and exact nature of a transaction. If the bargaining produces agreement on terms, the transaction takes place. Bargaining is an alternative pricing strategy to fixed prices. Optimally, if it costs the retailer nothing to engage and allow bargaining, they can deduce the buyer's willingness to spend. It allows for capturing more consumer surplus as it allows price discrimination, a process whereby a seller can charge a higher price to one buyer who is more eager. Haggling has largely disappeared in parts of the world where the cost to haggle exceeds the gain to retailers for most common retail items. However, for expensive goods sold to uninformed buyers such as automobiles, bargaining can remain commonplace.

In game theory, a **cooperative game** is a game with competition between groups of players ("coalitions") due to the possibility of external enforcement of cooperative behavior. Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing.

In game theory, the **best response** is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

In game theory, the **centipede game**, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

**Matching pennies** is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even keeps both pennies, so wins one from Odd. If the pennies do not match Odd keeps both pennies, so receives one from Even.

In game theory, a **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

An **extensive-form game** is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

In game theory, **folk theorems** are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

In game theory, a **repeated game** is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players; this impact is sometimes called his or her reputation. *Single stage game* or *single shot game* are names for non-repeated games.

In game theory, a **correlated equilibrium** is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

In game theory, a **subgame perfect equilibrium** is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game, no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium.

**Bargaining power** is the relative power of parties in a situation to exert influence over each other. If both parties are on an equal footing in a debate, then they will have equal bargaining power, such as in a perfectly competitive market, or between an evenly matched monopoly and monopsony.

**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 James J. O’Connor Distinguished Professor of Decision and Game Sciences at Northwestern University, 1975-2017, and currently is a Professor Emeritus of Managerial Economics and Decision Sciences.

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 was a mystery; thus, Rubinstein's solution is one of the most influential findings in game theory.

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

**Mertens stability** is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

**Resource 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 principle has been studied in various division problems.

The **Kalai–Smorodinsky (KS) bargaining solution** is a solution to the Bargaining problem. It was suggested by Ehud Kalai and Meir Smorodinsky, as an alternative to Nash's bargaining solution suggested 25 years earlier. The main difference between the two solutions is that the Nash solution satisfies independence of irrelevant alternatives while the KS solution satisfies monotonicity.

- ↑ Walker, Paul (2005). "History of Game Theory". Archived from the original on 2000-08-15. Retrieved 2008-05-03.
- ↑ Zeuthen, Frederik (1930).
*Problems of Monopoly and Economic Warfare*. - 1 2 Nash, John (1953-01-01). "Two-Person Cooperative Games".
*Econometrica*.**21**(1): 128–140. doi:10.2307/1906951. JSTOR 1906951. - 1 2 Rubinstein, Ariel (1982-01-01). "Perfect Equilibrium in a Bargaining Model".
*Econometrica*.**50**(1): 97–109. CiteSeerX 10.1.1.295.1434 . doi:10.2307/1912531. JSTOR 1912531. - ↑ Abhinay Muthoo "Bargaining Theory with Applications", Cambridge University Press, 1999.
- ↑ Nash, John (1950). "The Bargaining Problem".
*Econometrica*.**18**(2): 155–162. doi:10.2307/1907266. JSTOR 1907266. - ↑ Muthoo, Abhinay (1999).
*Bargaining theory with applications*. Cambridge University Press. - ↑ Kalai, Ehud & Smorodinsky, Meir (1975). "Other solutions to Nash's bargaining problem".
*Econometrica*.**43**(3): 513–518. doi:10.2307/1914280. JSTOR 1914280. - ↑ Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons" (PDF).
*Econometrica*.**45**(7): 1623–1630. doi:10.2307/1913954. JSTOR 1913954. - ↑ Schellenberg, James A. (1 January 1990). "'Solving' the Bargaining Problem" (PDF).
*Mid-American Review of Sociology*.**14**(1/2): 77–88. Retrieved 28 January 2017. - ↑ Felsenthal, D. S.; Diskin, A. (1982). "The Bargaining Problem Revisited: Minimum Utility Point, Restricted Monotonicity Axiom, and the Mean as an Estimate of Expected Utility".
*Journal of Conflict Resolution*.**26**(4): 664–691. doi:10.1177/0022002782026004005. - ↑ Binmore, Kenneth (1998).
*Game Theory and the Social Contract Volume 2: Just Playing*. Cambridge: MIT Press. ISBN 978-0-262-02444-0. - ↑ Binmore, Kenneth (2005).
*Natural Justice*. New York: Oxford University Press. ISBN 978-0-19-517811-1. - ↑ Gintis, H. (11 August 2016). "Behavioral ethics meets natural justice".
*Politics, Philosophy & Economics*.**5**(1): 5–32. doi:10.1177/1470594x06060617. - ↑ Osborne, Martin (1994).
*A Course in Game Theory*. MIT Press. ISBN 978-0-262-15041-5. - ↑ Roth, Alvin E.; Rothblum, Uriel G. (1982). "Risk Aversion and Nash's Solution for Bargaining Games with Risky Outcomes".
*Econometrica*.**50**(3): 639. doi:10.2307/1912605. JSTOR 1912605.

- Binmore, K.; Rubinstein, A.; Wolinsky, A. (1986). "The Nash Bargaining Solution in Economic Modelling".
*RAND Journal of Economics*.**17**(2): 176–188. doi:10.2307/2555382. JSTOR 2555382.

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