Cooperative bargaining

<h2 id="formal-description">Formal description</h2>
<p>A two-person bargain problem consists of:
</p>
<ul><li>A feasibility set 
  
    
      
        F
      
    
    {\displaystyle F}
  
, a closed subset of 
  
    
      
        
          
            R
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} ^{2}}
  
 that is often assumed to be convex, the elements of which are interpreted as agreements.</li>
<li>A disagreement, or threat, point 
  
    
      
        d
        =
        (
        
          d
          
            1
          
        
        ,
        
          d
          
            2
          
        
        )
      
    
    {\displaystyle d=(d_{1},d_{2})}
  
, where 
  
    
      
        
          d
          
            1
          
        
      
    
    {\displaystyle d_{1}}
  
 and 
  
    
      
        
          d
          
            2
          
        
      
    
    {\displaystyle d_{2}}
  
 are the respective payoffs to player 1 and player 2, which they are guaranteed to receive if they cannot come to a mutual agreement.</li></ul>
<p>The problem is nontrivial if agreements in 
  
    
      
        F
      
    
    {\displaystyle F}
  
 are better for both parties than the disagreement point. A solution to the bargaining problem selects an agreement 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 in 
  
    
      
        F
      
    
    {\displaystyle F}
  
.
</p>
<h3>Feasibility set</h3>
<p>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.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h3>Disagreement point</h3>
<p>The disagreement point 
  
    
      
        d
      
    
    {\displaystyle d}
  
 is the value the players can expect to receive if negotiations break down. This could be some <a href="/facts/Focal_point_(game_theory)/zfxTVs91">focal equilibrium</a> that both players could expect to play, or zero if no agreement is reached. 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 <i>Nash's variable threat game</i>.
</p>
<h2 id="nash-bargaining-game">Nash bargaining game</h2>
<p><a href="/facts/John_Forbes_Nash_Jr./wAg6sMSY">John Forbes Nash Jr.</a> came up with the Nash bargaining solution. It is the unique solution to a two-person bargaining problem that satisfies the axioms of <i>scale invariance</i>, <i>symmetry</i>, <i>efficiency</i>, and <i>independence of irrelevant alternatives</i>. According to Paul Walker,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Nash's bargaining solution was shown by <a href="/facts/John_Harsanyi/ujVSeu3x">John Harsanyi</a> to be the same as <a href="/facts/Zeuthen_Strategy/GQ2XIncc">Zeuthen</a>'s solution<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> of the bargaining problem.
</p><p>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.
</p><p>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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="equilibrium-analysis">Equilibrium analysis</h2>
<p>Strategies are represented in the Nash demand game by a pair (<i>x</i>, <i>y</i>). <i>x</i> and <i>y</i> are selected from the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [<i>d</i>, <i>z</i>], where <i>d</i> is the disagreement outcome and <i>z</i> is the total amount of good. If <i>x</i> + <i>y</i> is equal to or less than <i>z</i>, the first player receives <i>x</i> and the second <i>y</i>. Otherwise both get <i>d</i>; often 
  
    
      
        d
        =
        0
      
    
    {\displaystyle d=0}
  
.
</p><p>There are many <a href="/facts/Nash_equilibria/l74xzIHd">Nash equilibria</a> in the Nash demand game. Any <i>x</i> and <i>y</i> such that <i>x</i> + <i>y</i> = <i>z</i> 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 <i>x</i> or <i>y</i>. 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.
</p><p>In Rubinstein's alternating offers bargaining game,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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.
</p>
<h2 id="bargaining-solutions">Bargaining solutions</h2>
<p>Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.
</p>
<h3>Nash bargaining solution</h3>
<p><a href="/facts/John_Forbes_Nash_Jr./wAg6sMSY">John Forbes Nash Jr.</a> proposed that a solution should satisfy certain axioms:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<ol><li>Invariant to affine transformations or Invariant to equivalent utility representations</li>
<li><a href="/facts/Pareto_optimality/nMTdVPqy">Pareto optimality</a></li>
<li><a href="/facts/Independence_of_irrelevant_alternatives/E5xfFDsC">Independence of irrelevant alternatives</a></li>
<li><a href="/facts/Anonymity_(social_choice)/Kf9QndHL">Symmetry</a></li></ol>
<p>Nash proved that the solutions satisfying these axioms are exactly the points 
  
    
      
        (
        x
        ,
        y
        )
      
    
    {\displaystyle (x,y)}
  
 in 
  
    
      
        F
      
    
    {\displaystyle F}
  
 which maximize the following expression:
</p>

(
        u
        (
        x
        )
        −
        u
        (
        d
        )
        )
        (
        v
        (
        y
        )
        −
        v
        (
        d
        )
        )
      
    
    {\displaystyle (u(x)-u(d))(v(y)-v(d))}

<p>where <i>u</i> and <i>v</i> 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 
  
    
      
        (
        u
        (
        x
        )
        −
        u
        (
        d
        )
        )
        (
        v
        (
        y
        )
        −
        v
        (
        d
        )
        )
      
    
    {\displaystyle (u(x)-u(d))(v(y)-v(d))}
  
, where 
  
    
      
        u
        (
        d
        )
      
    
    {\displaystyle u(d)}
  
 and 
  
    
      
        v
        (
        d
        )
      
    
    {\displaystyle v(d)}
  
, are the <a href="/facts/Status_quo/kMsM2X2s">status quo</a> 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 <i>Nash product</i>. 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.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: 15–16 
</p>
<h3>Kalai–Smorodinsky bargaining solution</h3>
<p class="note">Main article: <a href="/facts/Kalai%E2%80%93Smorodinsky_bargaining_solution/JyNtcN6r">Kalai–Smorodinsky bargaining solution</a></p>
<p><a href="/facts/Independence_of_irrelevant_alternatives/E5xfFDsC">Independence of irrelevant alternatives</a> can be substituted with a <a href="/facts/Resource_monotonicity/yIvcN1Fh">resource monotonicity</a> axiom, as suggested by <a href="/facts/Ehud_Kalai/cAQ9rhny">Ehud Kalai</a> and Meir Smorodinsky.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> This leads to the <i>Kalai–Smorodinsky rule</i>, which selects the point which maintains the ratio of maximal gains. In other words, if we normalize the disagreement point to (0,0) and player 1 can receive a maximum of 
  
    
      
        
          g
          
            1
          
        
      
    
    {\displaystyle g_{1}}
  
 with player 2's help (and vice versa for 
  
    
      
        
          g
          
            2
          
        
      
    
    {\displaystyle g_{2}}
  
), then the Kalai–Smorodinsky bargaining solution would yield the point 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 on the Pareto frontier such that 
  
    
      
        
          ϕ
          
            1
          
        
        
          /
        
        
          ϕ
          
            2
          
        
        =
        
          g
          
            1
          
        
        
          /
        
        
          g
          
            2
          
        
      
    
    {\displaystyle \phi _{1}/\phi _{2}=g_{1}/g_{2}}
  
 .
</p>
<h3>Egalitarian bargaining solution</h3>
<p>The <a href="/facts/Egalitarian_rule/pvL7sd7m">egalitarian</a> bargaining solution, introduced by Ehud Kalai, is a third solution which drops the condition of scale invariance while including both the axiom of <a href="/facts/Independence_of_irrelevant_alternatives/E5xfFDsC">independence of irrelevant alternatives</a>, and the axiom of <a href="/facts/Resource_monotonicity/yIvcN1Fh">resource monotonicity</a>. 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 <a href="/facts/Egalitarian/ggyw0hYI">egalitarian</a> ideas of <a href="/facts/John_Rawls/Nqz88Ff0">John Rawls</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>
<h3>Comparison table</h3>
<table><tbody><tr><th>Name</th><th>Pareto-optimality</th><th>Symmetry</th><th>Scale-invariance</th><th>Irrelevant-independence</th><th>Resource-monotonicity</th><th>Principle</th></tr><tr><td><a href="/facts/Proportional-fair_rule/XrlfaYP4">Proportional-fair rule</a></td><td></td><td></td><td></td><td></td><td></td><td>Maximizing the <i>product</i> of surplus utilities</td></tr><tr><td><a href="/facts/Kalai%E2%80%93Smorodinsky_bargaining_solution/JyNtcN6r">Kalai–Smorodinsky rule</a></td><td></td><td></td><td></td><td></td><td></td><td>Equalizing the ratios of maximal gains</td></tr><tr><td><a href="/facts/Egalitarian_rule/pvL7sd7m">Egalitarian rule</a></td><td></td><td></td><td></td><td></td><td></td><td>Maximizing the <i>minimum</i> of surplus utilities</td></tr></tbody></table>
<h2 id="experimental-solutions">Experimental solutions</h2>
<p>A series of experimental studies<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> 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.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> 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.
</p>
<h2 id="applications">Applications</h2>
<p><a href="/facts/Kenneth_Binmore/0RJX9nw5">Kenneth Binmore</a> has used the Nash bargaining game to explain the emergence of human attitudes toward <a href="/facts/Distributive_justice/jFb4hAQ4">distributive justice</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> He primarily uses <a href="/facts/Evolutionary_game_theory/dPlL3umh">evolutionary game theory</a> to explain how individuals come to believe that proposing a 50–50 split is the only <a href="/facts/Justice/9mLpoA6B">just</a> solution to the Nash bargaining game. <a href="/facts/Herbert_Gintis/VqPPWe7Q">Herbert Gintis</a> supports a similar theory, holding that humans have evolved to a predisposition for <a href="/facts/Strong_reciprocity/4XKRH2p5">strong reciprocity</a> but do not necessarily make decisions based on direct consideration of utility.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a>
</p>
<h2 id="bargaining-solutions-and-risk-aversion">Bargaining solutions and risk-aversion</h2>
<p>Some economists have studied the effects of <a href="/facts/Risk_aversion/erarEwDf">risk aversion</a> 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.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a>: 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 <a href="/facts/Alvin_E._Roth/9s8ubcf2">Alvin E. Roth</a> and <a href="/facts/Uriel_Rothblum/SP37NWgQ">Uriel Rothblum</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bargaining/KcUxrewP">Bargaining</a></li>
<li><a href="/facts/Core_(game_theory)/79uDsDct">Core (game theory)</a></li>
<li><a href="/facts/Rubinstein_bargaining_model/mb97Aw95">Rubinstein bargaining model</a></li>
<li><a href="/facts/Sequential_bargaining/P4BWynoM">Sequential bargaining</a></li>
<li><a href="/facts/Nash_equilibrium/l74xzIHd">Nash equilibrium</a></li>
<li><a href="/facts/Ultimatum_game/LxV4CLRJ">Ultimatum game</a></li></ul>

<ul><li>Binmore, K.; Rubinstein, A.; Wolinsky, A. (1986). "The Nash Bargaining Solution in Economic Modelling". <i><a href="/facts/RAND_Journal_of_Economics/TkFsr1u2">RAND Journal of Economics</a></i>. 17 (2): 176–188. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2555382">10.2307/2555382</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2555382">2555382</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.cse.iitd.ernet.in/~rahul/cs905/lecture15/">Nash bargaining solutions</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Thomson, William (1994-01-01), "Chapter 35 Cooperative models of bargaining", Handbook of Game Theory with Economic Applications, vol. 2, Elsevier, pp. 1237–1284, retrieved 2021-02-05 <a href="https://www.sciencedirect.com/science/article/pii/S1574000505800670" target="_blank">https://www.sciencedirect.com/science/article/pii/S1574000505800670</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Nash, John (1953-01-01). "Two-Person Cooperative Games". Econometrica. 21 (1): 128–140. doi:10.2307/1906951. JSTOR 1906951. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Walker, Paul (2005). "History of Game Theory". Archived from the original on 2000-08-15. Retrieved 2008-05-03. <a href="https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html#ref94" target="_blank">https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html#ref94</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Zeuthen, Frederik (1930). Problems of Monopoly and Economic Warfare. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Nash, John (1953-01-01). "Two-Person Cooperative Games". Econometrica. 21 (1): 128–140. doi:10.2307/1906951. JSTOR 1906951. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>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. S2CID 14827857. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Nash, John (1950). "The Bargaining Problem". Econometrica. 18 (2): 155–162. doi:10.2307/1907266. JSTOR 1907266. S2CID 153422092. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Muthoo, Abhinay (1999). Bargaining theory with applications. Cambridge University Press. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Kalai, Ehud & Smorodinsky, Meir (1975). "Other solutions to Nash's bargaining problem". Econometrica. 43 (3): 513–518. doi:10.2307/1914280. JSTOR 1914280. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Kalai, Ehud (1977). "Proportional solutions to bargaining situations: Intertemporal utility comparisons" (PDF). Econometrica. 45 (7): 1623–1630. doi:10.2307/1913954. JSTOR 1913954. <a href="http://www.kellogg.northwestern.edu/research/math/papers/179.pdf" target="_blank">http://www.kellogg.northwestern.edu/research/math/papers/179.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>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. <a href="https://kuscholarworks.ku.edu/bitstream/handle/1808/5046/MARSV14N1-2A7.pdf?sequence=1&isAllowed=y" target="_blank">https://kuscholarworks.ku.edu/bitstream/handle/1808/5046/MARSV14N1-2A7.pdf?sequence=1&isAllowed=y</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>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. S2CID 154770122. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Binmore, Kenneth (1998). Game Theory and the Social Contract Volume 2: Just Playing. Cambridge: MIT Press. ISBN 978-0-262-02444-0. <a href="978-0-262-02444-0" target="_blank">978-0-262-02444-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Binmore, Kenneth (2005). Natural Justice. New York: Oxford University Press. ISBN 978-0-19-517811-1. <a href="978-0-19-517811-1" target="_blank">978-0-19-517811-1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Gintis, H. (11 August 2016). "Behavioral ethics meets natural justice". Politics, Philosophy & Economics. 5 (1): 5–32. doi:10.1177/1470594x06060617. S2CID 19601647. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Osborne, Martin (1994). A Course in Game Theory. MIT Press. ISBN 978-0-262-15041-5. <a href="978-0-262-15041-5" target="_blank">978-0-262-15041-5</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
</ol>

Cooperative bargaining open-in-new

Cooperative bargaining