Bargaining problem

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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

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]

Formal description

A two-person bargain problem consists of:

  • A feasibility set [math]\displaystyle{ F }[/math], a closed subset of [math]\displaystyle{ \mathbb{R}^2 }[/math] that is often assumed to be convex, the elements of which are interpreted as agreements. [math]\displaystyle{ F }[/math] 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 [math]\displaystyle{ d=(d_1, d_2) }[/math], where [math]\displaystyle{ d_1 }[/math] and [math]\displaystyle{ d_2 }[/math] 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 [math]\displaystyle{ F }[/math] are better for both parties than the disagreement point. A solution to the bargaining problem selects an agreement [math]\displaystyle{ \phi }[/math] in [math]\displaystyle{ F }[/math].

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 the agents that are bargaining.[3]

Disagreement point

The disagreement point [math]\displaystyle{ d }[/math] 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.

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 [math]\displaystyle{ d=0 }[/math].

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.

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 Nash proposed[6] that a solution should satisfy certain axioms:

  1. Invariant to affine transformations or Invariant to equivalent utility representations
  2. Pareto optimality
  3. Independence of irrelevant alternatives
  4. Symmetry

Nash proved that the solutions satisfying these axioms are exactly the points [math]\displaystyle{ (x,y) }[/math] in [math]\displaystyle{ F }[/math] which maximize the following expression:

[math]\displaystyle{ (u(x)-u(d))(v(y)-v(d)) }[/math]

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 [math]\displaystyle{ (u(x)-u(d))(v(y)-v(d)) }[/math], where [math]\displaystyle{ u(d) }[/math] and [math]\displaystyle{ v(d) }[/math], 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

Kalai–Smorodinsky bargaining solution

Main page: Kalai–Smorodinsky bargaining solution

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 [math]\displaystyle{ g_1 }[/math] with player 2's help (and vice versa for [math]\displaystyle{ g_2 }[/math]), then the Kalai–Smorodinsky bargaining solution would yield the point [math]\displaystyle{ \phi }[/math] on the Pareto frontier such that [math]\displaystyle{ \phi_1 / \phi_2 = g_1 / g_2 }[/math] .

Egalitarian bargaining solution

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.

Comparison table

Name Pareto-optimality Symmetry Scale-invariance Irrelevant-independence Resource-monotonicity Principle
Nash (1950) Yes Yes Yes Yes No Maximizing the product of surplus utilities
Kalai-Smorodinsky (1975) Yes Yes Yes No Yes Equalizing the ratios of maximal gains
Kalai (1977) Yes Yes No Yes Yes Maximizing the minimum of surplus utilities

Experimental solutions

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.

Applications

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]

Bargaining solutions and risk-aversion

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]

See also

References

  1. ↑ Walker, Paul (2005). "History of Game Theory". http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm#ref94. 
  2. ↑ Zeuthen, Frederik (1930). Problems of Monopoly and Economic Warfare. 
  3. ↑ 3.0 3.1 Nash, John (1953-01-01). "Two-Person Cooperative Games". Econometrica 21 (1): 128–140. doi:10.2307/1906951. 
  4. ↑ 4.0 4.1 Rubinstein, Ariel (1982-01-01). "Perfect Equilibrium in a Bargaining Model". Econometrica 50 (1): 97–109. doi:10.2307/1912531. 
  5. ↑ Abhinay Muthoo "Bargaining Theory with Applications", Cambridge University Press, 1999.
  6. ↑ Nash, John (1950). "The Bargaining Problem". Econometrica 18 (2): 155–162. doi:10.2307/1907266. 
  7. ↑ Muthoo, Abhinay (1999). Bargaining theory with applications. Cambridge University Press. 
  8. ↑ Kalai, Ehud; Smorodinsky, Meir (1975). "Other solutions to Nash's bargaining problem". Econometrica 43 (3): 513–518. doi:10.2307/1914280. 
  9. ↑ Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons". Econometrica 45 (7): 1623–1630. doi:10.2307/1913954. http://www.kellogg.northwestern.edu/research/math/papers/179.pdf. 
  10. ↑ Schellenberg, James A. (1 January 1990). "'Solving' the Bargaining Problem". Mid-American Review of Sociology 14 (1/2): 77–88. https://kuscholarworks.ku.edu/bitstream/handle/1808/5046/MARSV14N1-2A7.pdf?sequence=1&isAllowed=y. Retrieved 28 January 2017. 
  11. ↑ 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. 
  12. ↑ Binmore, Kenneth (1998). Game Theory and the Social Contract Volume 2: Just Playing. Cambridge: MIT Press. ISBN 978-0-262-02444-0. 
  13. ↑ Binmore, Kenneth (2005). Natural Justice. New York: Oxford University Press. ISBN 978-0-19-517811-1. 
  14. ↑ Gintis, H. (11 August 2016). "Behavioral ethics meets natural justice" (in en). Politics, Philosophy & Economics 5 (1): 5–32. doi:10.1177/1470594x06060617. 
  15. ↑ Osborne, Martin (1994). A Course in Game Theory. MIT Press. ISBN 978-0-262-15041-5. 
  16. ↑ 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. 
  • 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. 

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