¹ Nash, John, “The Bargaining Problem,” Econometrica 18 (1950): 155-62CrossRefGoogle Scholar and “Two-Person Cooperative Games,” Econometrica 21 (1953): 128–40.

² Braithwaite, Richard, Theory of Games as a Tool for the Moral Philosopher (Bristol, UK: Thoemmes Press, [1955] 1994).Google Scholar

³ Aristotle, Politics 1282b18-22. See also Politics 1280a8-30 and Nicomachean Ethics 1131a-1131b22.

⁴ Following the common practice in the game-theoretic literature, in a 2-agent game at each outcome determined by the agents’ individual strategy choices the row (column) agent’s payoff is the first (second) component of the corresponding payoff vector. For example, at the (G, M) outcome of the Figure 1 game Claudia’s payoff is 1 and Laura’s payoff is 0.

⁵ In this and the other expanded demand games discussed in this essay, the payoffs are derived from the basis game payoffs as follows: If ${u_1}\left( {{s_1},{s_2}} \right)$ denotes Agent 1’s (Claudia’s) payoff at the strategy profile u ₂(s ₁, s ₂) in the basis game, then in the expanded demand game Agent 1’s payoff at the claim profile (x ₁, x ₂) is defined as

$${u_1}\left( {{x_1},{x_2}} \right) = \left\{ {\matrix{ {{x_1} \cdot {u_1}\left( {G,M} \right) + {x_2} \cdot {u_1}\left( {M,G} \right) - \left( {1 - {x_1} - {x_2}} \right) \cdot {u_1}\left( {M,M} \right){\rm{\ if\ }}{x_1} + {x_2} \le 1} \cr {0{\rm{\ otherwise}}} \cr} } \right.\,.$$

Agent 2’s (Laura’s) payoffs are similarly defined.

⁶ This alternation scheme between (G, M) and (M, G) is a correlated equilibrium. Robert Aumann formalized the correlated equilibrium concept, which generalizes the Nash equilibrium concept, in the essays “Subjectivity and Correlation in Randomized Strategies,” Journal of Mathematical Economics 1 (1974): 67–96 and “Correlated Equilibrium as An Expression of Bayesian Rationality,” Econometrica 55 (1987): 1–18.

Lewis, David K. presented an early game-theoretic analysis of convention in terms of the Nash equilibrium concept in Convention: A Philosophical Study (Cambridge, MA: Harvard University Press, 1969).Google Scholar I later proposed a correlated equilibrium analysis of convention that generalizes Lewis’s analysis. See Peter Vanderschraaf, “Knowledge, Equilibrium and Convention,” Erkenntnis 49 (1998): 337–69, and Strategic Justice: Convention and Problems of Balancing Divergent Interests (New York: Oxford University Press, forthcoming), chap. 2.

⁷ A proposition A is common knowledge for a group of agents if each group member knows A, each group member knows that each group member knows A, and so on, ad infinitum. Lewis gives an early account of common knowledge in Lewis, Convention, 52-60.

⁸ Braithwaite, Theory of Games as a Tool for the Moral Philosopher, 4, 7.

⁹ In some presentations of axiomatic bargaining theory, the feasible set is defined as a larger comprehensive set. The comprehensive set extension of the compatible claim payoff vector set reflects an additional assumption that each agent is free to destroy without cost any of the good she might receive from a division defined by compatible claims.

¹⁰ A simple example of a trivial bargaining problem is a 2-agent demand game where each Agent i can claim any desired fraction x _i of a good at stake, same as in the Chocolate Cake and Braithwaite problems, but where ${u_i}\left( {{x_1},{x_2}} \right) = 0$ for any set of claims, compatible or incompatible. This example summarizes a situation where neither agent happens to derive any positive payoff from receiving any amount of the good at stake.

¹¹ Nash, “Two-Person Cooperative Games,” 128–29.

¹² The Nash product of the Braithwaite Demand Game is $x \cdot \left( {{5 \over 3} - {{14x} \over 9}} \right)$, which is maximized at $x = {{15} \over {28}}$, which in turn defines the Pareto frontier point $\left( {{{15} \over {28}},{{17} \over {18}}} \right) \approx \left( {0.536,0.944} \right)$. Solving $z \cdot \left( {{1 \over 2},1} \right) + \left( {1 - z} \right) \cdot \left( {1,{2 \over 9}} \right) = \left( {{{15} \over {28}},{{17} \over {18}}} \right)$ yields $z = {1 \over {14}}$, so at the Nash solution Claudia claims ${1 \over {14}}$ and Laura claims ${{13} \over {14}}$.

¹³ See Roth, Alvin, Axiomatic Models of Bargaining (Berlin: Springer Verlag, 1979)CrossRefGoogle Scholar and Thomson, William and Lensberg, Terje, Axiomatic Theory of Bargaining With a Variable Number of Agents (Cambridge: Cambridge University Press, 1989)CrossRefGoogle Scholar for fine surveys of axiomatic solution concepts for bargaining problems.

¹⁴ Raiffa, Howard, “Arbitration Schemes for Generalized Two-Person Games,” in Contributions to the Theory of Games, vol. 2, ed. Kuhn, H. and Tucker, A. W. (Princeton, NJ: Princeton University Press, 1953), 361–87.Google Scholar

¹⁵ Braithwaite’s solution differs from Raiffa’s egalitarian solution in that Braithwaite adopts a different scaling of the payoffs. Consequently, at Braithwaite’s egalitarian solution the agents follow (G, M) for ${{16} \over {43}}$ of the time and (M, G) for ${{27} \over {43}}$ of the time.

¹⁶ Kalai, Ehud and Smorodinsky, Meir, “Other Solutions to Nash’s Bargaining Problem,” Econometrica 16 (1975): 29–56.Google Scholar

¹⁷ Gauthier, David, Morals by Agreement (Oxford: Clarendon Press, 1986), chap. 5.Google Scholar Gauthier presented an earlier defense of minimax relative concession in “Rational Co-Operation,” Nous 8 (1974): 53–65.

¹⁸ In more recent work, Gauthier endorses a closely related principle of maximin proportionate gain that he discussed in Morals by Agreement, 14–15, 154–55, but now defends using arguments inspired by Rubenstein’s analysis of strategic bargaining. See Gauthier, “Twenty-Five On,” Ethics 124 (2012): 601–624.

¹⁹ Raiffa’s and Braithwaite’s alternate egalitarian solutions of the Braithwaite Demand Game illustrate how this solution concept can fail to satisfy scale invariance.

²⁰ John Thrasher gives a fine critique of the role of the symmetry axiom in bargaining theory in “Uniqueness and Symmetry in Bargaining Theories of Justice,” Philosophical Studies 167 (2014): 683–99.

²¹ Ingolf Stähl, Bargaining Theory (Stockholm: Economic Research Institute, 1972) and Rubinstein, Ariel, “Perfect Equilibrium in a Bargaining Model,” Econometrica 50 (1982): 97–109.CrossRefGoogle Scholar

²² Martin J. Osborne and Ariel Rubinstein show by example that a 3-agent extension of the 2-agent Rubinstein model can converge to any outcome of the Pareto frontier in Bargaining and Markets (San Diego, CA: Academic Press Inc., 1990), 63–65. Osborne and Rubinstein credit the example and the 3-agent extension of the Rubinstein model to Avner Shaked.

²³ Van Damme, Eric, Selten, Reinhard and Winter, Eyal showed this in “Alternating Bid Bargaining with a Smallest Money Unit,” Games and Economic Behavior 2 (1990): 188–201.CrossRefGoogle Scholar

²⁴ Duncan Luce, R. and Raiffa, Howard, Games and Decisions: Introduction and Critical Survey (New York: John Wiley and Sons, 1957), 105Google Scholar; and John Nash, “Appendix: Motivation and Interpretation,” reprinted in Essays on Game Theory by John Nash, ed. Ken Binmore (Cheltenham, UK: Edward Elgar, [1951] 1996), 32–33. Nash’s appendix was to his doctoral thesis.

²⁵ Schelling, Thomas, The Strategy of Conflict (Cambridge, MA: Harvard University Press, 1960), 57.Google Scholar Schelling develops his views on coordination and focal points primarily in chapters 2 and 3.

²⁶ See especially A Treatise of Human Nature, Bk. III, Pt. I, Secs. I-3 and An Enquiry Concerning the Principles of Morals, Sec. III, Pt. II.

²⁷ See Roth, Alvin, “Bargaining Experiments,” in Kagel, John H. and Roth, Alvin E., eds. Handbook of Experimental Economics (Princeton, NJ: Princeton University Press, 1995), 253–348,Google Scholar and Camerer, Colin, Behavioral Game Theory: Experiments in Strategic Interaction (Princeton, NJ: Princeton University Press, 2003), chap. 4.Google Scholar

²⁸ See especially Schelling, The Strategy of Conflict, 54–58; and Hume, A Treatise of Human Nature Bk. III, Pt. II, Sec. III: 4, n. 1 and Bk. III, Pt. II, Sec. IV: 1-2 and An Enquiry Concerning the Principles of Morals, Sec. III, Pt. II, 35–37.

²⁹ Carnap’s final system of inductive logic was published posthumously as “A Basic System of Inductive Logic, Part 2,” in Studies in Inductive Logic and Probability, vol. II, ed. Richard Jeffrey (Berkeley, CA: University of California Press, 1980), 7–155.

³⁰ See Cheung, Yin-Wong and Friedman, Daniel, “Individual Learning in Normal Form Games: Some Laboratory Results,” Games and Economic Behavior 19 (1997): 46–76.CrossRefGoogle Scholar Camerer, Behavioral Game Theory, 283-95 summarizes the results of these and related experimental studies.

³¹ For a proof of these results see Proposition 2.1 of Drew Fudenberg and David K. Levine, The Theory of Learning in Games (Cambridge, MA: MIT Press, 1998), 33.

³² Gauthier, Morals by Agreement, 147–48.

³³ Gauthier uses this game partly to illustrate how the Nash and Kalai-Smorodinsky solutions can differ without mentioning the egalitarian solution.

³⁴ Luce and Raiffa, Games and Decisions, 139 introduced this game as part of their critical discussion of alternative procedures for analyzing bargaining problems. The name “False Mirror” is my own.

³⁵ Again, Braithwaite’s use of a payoff scale different from the Raiffa scale used in the Figure 2 game produces a solution different from the egalitarian solution based on the Figure 2 game. See note 15.

³⁶ Harsanyi, John “Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility,” Journal of Political Economy 63 (1955): 309–32 and Rational Behavior and Bargaining Equilibrium in Games and Social Situations (Cambridge: Cambridge University Press, 1977) chap. 4.Google Scholar However, it is important to note that Harsanyi, Rational Behavior and Bargaining Equilibrium in Games and Social Situations, sect. 9.11 states that while he believes interpersonal utility comparisons are crucial in making moral value judgments, he is much more doubtful that they should play a direct role in the solution of a bargaining problem. Unlike Raiffa and Braithwaite, Harsanyi does not consider the bargaining problem a proper tool for analyzing principles of fair division.

³⁷ Ken Binmore has produced a particularly sophisticated theory of interpersonal utility comparisons based upon empathetic preferences. Binmore develops elements of his theory in various writings, but gives his most comprehensive presentation in Ken Binmore, Game Theory and the Social Contract Volume I: Playing Fair (Cambridge, MA: MIT Press, 1994), chap. 4 and Game Theory and the Social Contract Volume II: Just Playing (Cambridge, MA: MIT Press, 1998), chap. 2.

³⁸ Luce and Raiffa, Games and Decisions, 131–32.

³⁹ Aristotle, Politics 1280a11-19, Nicomachean Ethics 1131a20-24. In his fine reconstruction of Aristotle’s account of distributive justice, David Keyt interprets Aristotle as maintaining that people tend to agree upon how to value shares of the good at stake, and tend to disagree mainly over which criteria are relevant criteria of worth. See David Keyt, “Aristotle’s Theory of Distributive Justice,” in A Companion to Aristotle’s Politics, ed. David Keyt and Fred D. Miller, Jr. (Oxford: Basil Blackwell, 1991), 242.

⁴⁰ Aristotle, Nicomachean Ethics 1131a25-28, Politics 1280a18-19.

⁴¹ For the first version of Aristotle’s formula given here, which reconstructs Aristotle’s final formulation in Nicomachean Ethics 1131b4-10, I draw upon Keyt, “Aristotle’s Theory of Distributive Justice,” 241-42.

⁴² Binmore, Game Theory and the Social Contract, 397-99, gives a similar argument for the equivalence of the Aristotelian proportionality rule in division problems and the egalitarian solution of the bargaining problem.

⁴³ The results of these studies are summarized in Roth, “Bargaining Experiements,” Camerer, Behavioral Game Theory, chap. 4, and Ken Binmore, Does Game Theory Work?: The Bargaining Challenge (Cambridge, MA: MIT Press, 2007), chap. 2.

⁴⁴ See especially Samuelson, Larry, “Does Evolution Eliminate Dominated Strategies?” in Frontiers of Game Theory, ed. Binmore, Ken, Kirman, Alan, and Tani, Piero (Cambridge, MA: MIT Press, 1993), 213–35,Google Scholar Samuelson, Evolutionary Games and Equilibrium Selection (Cambridge, MA: MIT Press, 1997), Brian Skyrms, Evolution of the Social Contract, 2nd ed. (Cambridge: Cambridge University Press, 2014 [1996]), chaps. 1–2, Harms, William, “Evolution and Ultimatum Bargaining,” Theory and Decision 42 (1997): 147–75,CrossRefGoogle Scholar and Wagner, Elliot, “Evolving to Divide the Fruits of Cooperation,” Philosophy of Science 79 (2012): 81–94.CrossRefGoogle Scholar

⁴⁵ For example, Skyrms, Evolution of the Social Contract, 107–8, applies the replicator dynamic to a Truncated Chocolate Cake game where the Nash and Kalai-Smorodinsky solutions differ in order to test the attracting power of the Nash solution. In this game the Nash solution is the same as the egalitarian solution, which Skyrms does not directly discuss.

⁴⁶ See especially Peyton Young, H., “An Evolutionary Model of Bargaining,” Journal of Economic Theory 59 (1993): 145–68,CrossRefGoogle Scholar Individual Strategy and Social Structure: An Evolutionary Theory of Institutions (Princeton, NJ: Princeton University Press, 1998), chaps. 8–9, Ken Binmore, Larry Samuelson, H. Peyton Young, “Equilibrium Selection in Bargaining Models,” Games and Economic Behavior 45 (2003): 296–328, and Jack Robles, “Evolution, Bargaining, and Time Preferences,” Economic Theory 35 (2008): 19–36.

⁴⁷ Sugden, Robert raises the same objection to models of equilibrium selection based on the adaptive learning dynamic in The Economics of Rights, Co-operation and Welfare, 2nd ed. (Houndsmills, Basingstoke, Hampshirer, and New York: Palgrave MacMillan, 2004 [1986]), 203–4.Google Scholar

The adaptive learning dynamic reaches its most persistent long term limits so slowly because this dynamic sets severe limits on the information of the history of interaction available to agents and incorporates “noise” in the form of independent random errors. Young, Individual Strategy and Social Structure, chaps. 8–9, gives a fine summary of the mechanics of this dynamic applied to bargaining problems.

⁴⁸ I constructed and ran these simulations in MatLab 8.

⁴⁹ See note 31.