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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,<ref>{{cite web |last=Walker |first=Paul |year=2005 |title=History of Game Theory |url=http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm#ref94 |access-date=2008-05-03 |archive-url=https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html#ref94 |archive-date=2000-08-15 |url-status=dead }}</ref> Nash's bargaining solution was shown by [[Biography:John Harsanyi|John Harsanyi]] to be the same as Zeuthen's solution<ref>{{cite book|last=Zeuthen|first= Frederik|title=Problems of Monopoly and Economic Warfare|year=1930}}</ref> 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.<ref name=":0">{{Cite journal|last=Nash|first=John|date=1953-01-01|title=Two-Person Cooperative Games|jstor=1906951|journal=Econometrica|volume=21|issue=1|pages=128–140|doi=10.2307/1906951}}</ref>

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.<ref name="Rubinstein 97–109">{{Cite journal|last=Rubinstein|first=Ariel|date=1982-01-01|title=Perfect Equilibrium in a Bargaining Model|jstor=1912531|journal=Econometrica|volume=50|issue=1|pages=97–109|doi=10.2307/1912531|citeseerx=10.1.1.295.1434}}</ref> 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.<ref>Abhinay Muthoo "[http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521576474 Bargaining Theory with Applications]", [http://www.cambridge.org/ Cambridge University Press], 1999.</ref>

== Formal description ==

A two-person bargain problem consists of:
* A feasibility set <math>F</math>, a closed subset of <math>\mathbb{R}^2</math> that is often assumed to be convex, the elements of which are interpreted as agreements. <math>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>d=(d_1, d_2)</math>, where <math>d_1</math> and <math>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>F</math> are better for both parties than the disagreement point. A solution to the bargaining problem selects an agreement <math>\phi</math> in <math>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.<ref name=":0" />

=== Disagreement point ===

The disagreement point <math>d</math> is the value the players can expect to receive if negotiations break down. This could be some [[Software:Focal point (game theory)|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 (mathematics)|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>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,<ref name="Rubinstein 97–109"/> 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 ===

[[Biography:John Forbes Nash Jr.|John Nash]] proposed<ref>{{cite journal |last=Nash |first=John |year=1950 |title=The Bargaining Problem |journal=Econometrica |volume=18 |issue=2 |pages=155–162 |doi= 10.2307/1907266|jstor=1907266 }}</ref> that a solution should satisfy certain axioms:

#Invariant to affine transformations or Invariant to equivalent utility representations
#Pareto optimality
#[[Social:Independence of irrelevant alternatives|Independence of irrelevant alternatives]]
#[[Symmetry]]

Nash proved that the solutions satisfying these axioms are exactly the points <math>(x,y)</math> in <math>F</math> which maximize the following expression:
::<math>(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>(u(x)-u(d))(v(y)-v(d))</math>, where <math>u(d)</math> and <math>v(d)</math>, are the [[Philosophy:Status quo|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.<ref>{{cite book|last=Muthoo|first=Abhinay|year=1999|title=Bargaining theory with applications|publisher=Cambridge University Press}}</ref>{{rp|15–16}}

=== Kalai–Smorodinsky bargaining solution ===
{{Main|Kalai–Smorodinsky bargaining solution}}
Independence of Irrelevant Alternatives can be substituted with a [[Social:Resource monotonicity|Resource monotonicity]] axiom. This was demonstrated by [[Biography:Ehud Kalai|Ehud Kalai]] and Meir Smorodinsky.<ref>{{cite journal |last=Kalai |first=Ehud |lastauthoramp=yes |first2=Meir |last2=Smorodinsky |year=1975 |title=Other solutions to Nash's bargaining problem |journal=Econometrica |volume=43 |issue=3 |pages=513–518 |doi= 10.2307/1914280|jstor=1914280 }}</ref> 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>g_1</math>  with player 2's help (and vice versa for <math>g_2</math>), then the Kalai–Smorodinsky bargaining solution would yield the point <math>\phi</math> on the Pareto frontier such that <math>\phi_1 / \phi_2 = g_1 / g_2  </math> .

=== Egalitarian bargaining solution ===
The egalitarian bargaining solution, introduced by Ehud Kalai,<ref>{{cite journal |last=Kalai |first=Ehud |year=1977 |title=Proportional solutions to bargaining situations: Intertemporal utility comparisons |journal=[[Finance:Econometrica|Econometrica]] |volume=45 |issue=7 |pages=1623–1630 |doi= 10.2307/1913954|jstor=1913954 |url=http://www.kellogg.northwestern.edu/research/math/papers/179.pdf }}</ref> is a third solution which drops the condition of scale invariance while including both the axiom of [[Social:Independence of irrelevant alternatives|Independence of irrelevant alternatives]], and the axiom of [[Social:Resource monotonicity|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 [[Biography:John Rawls|John Rawls]].

=== Comparison table ===
{| class="wikitable sortable"
|-
! Name !! Pareto-optimality !! Symmetry !! Scale-invariance !! Irrelevant-independence !! Resource-monotonicity !! Principle
|-
| Nash (1950) || {{ya}} || {{ya}} || {{ya}} || {{ya}} || {{na}} || Maximizing the ''product'' of surplus utilities
|-
| Kalai-Smorodinsky (1975) || {{ya}} || {{ya}} || {{ya}} || {{na}} || {{ya}} ||  Equalizing the ratios of maximal gains 
|-
| Kalai (1977) || {{ya}} || {{ya}} || {{na}} || {{ya}} || {{ya}} || Maximizing the ''minimum'' of surplus utilities
|}

== Experimental solutions ==

A series of experimental studies<ref>{{cite journal|last1=Schellenberg|first1=James A.|title='Solving' the Bargaining Problem|journal=Mid-American Review of Sociology|date=1 January 1990|volume=14|issue=1/2|pages=77–88|url=https://kuscholarworks.ku.edu/bitstream/handle/1808/5046/MARSV14N1-2A7.pdf?sequence=1&isAllowed=y|accessdate=28 January 2017}}</ref> 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.<ref>{{cite journal|last1=Felsenthal|first1=D. S.|last2=Diskin|first2=A.|title=The Bargaining Problem Revisited: Minimum Utility Point, Restricted Monotonicity Axiom, and the Mean as an Estimate of Expected Utility|journal=Journal of Conflict Resolution|volume=26|issue=4|pages=664–691|doi=10.1177/0022002782026004005|year=1982}}</ref> 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 ==

[[Biography:Kenneth Binmore|Kenneth Binmore]] has used the Nash bargaining game to explain the emergence of human attitudes toward [[Philosophy:Distributive justice|distributive justice]].<ref>{{cite book |last=Binmore |first=Kenneth |year=1998 |title=Game Theory and the Social Contract Volume 2: Just Playing |location=Cambridge |publisher=MIT Press |isbn=978-0-262-02444-0 }}</ref><ref>{{cite book |last=Binmore |first=Kenneth |year=2005 |title=Natural Justice |location=New York |publisher=Oxford University Press |isbn=978-0-19-517811-1 }}</ref> He primarily uses [[Evolutionary game theory|evolutionary game theory]] to explain how individuals come to believe that proposing a 50–50 split is the only [[Philosophy:Justice|just]] solution to the Nash bargaining game. [[Biography:Herbert Gintis|Herbert Gintis]] supports a similar theory, holding that humans have evolved to a predisposition for [[Biology:Strong reciprocity|strong reciprocity]] but do not necessarily make decisions based on direct consideration of utility.<ref>{{cite journal|last1=Gintis|first1=H.|title=Behavioral ethics meets natural justice|journal=Politics, Philosophy & Economics|date=11 August 2016|volume=5|issue=1|pages=5–32|doi=10.1177/1470594x06060617|language=en}}</ref>

== Bargaining solutions and risk-aversion ==
Some economists have studied the effects of [[Risk aversion|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.<ref name="OR94">{{cite book|title=A Course in Game Theory|publisher=MIT Press|year=1994|isbn=978-0-262-15041-5|first1=Martin|last1=Osborne}}</ref>{{rp|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 [[Biography:Alvin E. Roth|Alvin E. Roth]] and Uriel Rothblum<ref>{{cite journal|doi=10.2307/1912605|jstor=1912605|title=Risk Aversion and Nash's Solution for Bargaining Games with Risky Outcomes|journal=Econometrica|volume=50|issue=3|pages=639|year=1982|last1=Roth|first1=Alvin E.|last2=Rothblum|first2=Uriel G.}}</ref>

== See also ==
*[[Bargaining]]
*[[Rubinstein bargaining model]]
*[[Nash equilibrium]]
*Ultimatum game

== References ==
{{reflist|30em}}
*{{cite journal |last=Binmore |first=K. |last2=Rubinstein |first2=A. |last3=Wolinsky |first3=A. |year=1986 |title=The Nash Bargaining Solution in Economic Modelling |journal=RAND Journal of Economics |volume=17 |issue= 2|pages=176–188 |doi= 10.2307/2555382|jstor=2555382 }}

== External links ==
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*[http://www.economics.laurentian.ca/Strategic_Think.27/Modules/Course_Schedule.98/Bargaining/Nash%20solution/nash_solution.htm D. Robinson's page "The Nash Solution for a Bargaining Game"]
*[http://www.economics.laurentian.ca/Strategic_Think.27/Modules/Course_Schedule.98/Bargaining/variable%20threat%20game.htm "variable threat games"]

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* [http://www.cse.iitd.ernet.in/~rahul/cs905/lecture15/ Nash bargaining solutions]

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[[Category:Cooperative games]]

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