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Nash Equilibrium

A nash equilibrium for a game is a set of actions from which no agent can unilaterally profitably deviate (Osborne & Rubinstein, 1994, p. 14).

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Notes

Game theory is supposed to explain why things happen:

‘Many events and outcomes prompt us to ask: Why did that happen? […] For example, cutthroat competition in business is the result of the rivals being trapped in a prisoners’ dilemma’ (Dixit, Skeath, & Reiley, 2014, p. 36).

This section introduces two notions that are involved in giving such explanations, dominance and Nash equilibrium.

If you understand these notions and can apply them, you can do game theory.

Nash Equilibrium

A Nash equilibrium for a game is a set of actions (sometimes called a ‘strategy’) from which no agent can unilaterally profitably deviate.

Why equilibrium?:

‘equilibrium […] simply means that each player is using the strategy that is the best response to the strategies of the other players’ (Dixit et al., 2014, pp. 32–3)

Although not covered in this section, there is some interesting research on other ways of specifying a ‘best response’ (Misyak & Chater, 2014; Misyak & Chater, 2014b). Why might you want to do so? Potential motives arise in Consequences and Applications of Game Theory and What Is Team Reasoning?.

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Glossary

dominance : An action (or strategy) _strictly dominates_ another if it ensures better outcomes for its player no matter what other players choose. (See also weak dominance.)
game theory : This term is used for any version of the theory based on the ideas of Neumann et al. (1953) and presented in any of the standard textbooks including. Hargreaves-Heap & Varoufakis (2004); Osborne & Rubinstein (1994); Tadelis (2013); Rasmusen (2007).
Nash equilibrium : a profile of actions (sometimes called a ‘strategy’) from which no agent can unilaterally profitably deviate.
noncooperative game : ‘Games in which joint-action agreements are enforceable are called _cooperative_ games; those in which such enforcement is not possible, and individual participants must be allowed to act in their own interests, are called _noncooperative_ games’ (Dixit et al., 2014, p. 26).
strict dominance : In game theory, one action _strictly dominates_ another action if the first action guarantees its player higher payoffs than the second action regardless of what other players choose to do. (See Definition 59.2 in Osborne & Rubinstein, 1994, p. 59 for a more general definition.)
weak dominance : In game theory, one action _weakly dominates_ another action if the first action guarantees its player payoffs at least as good as the other action and potentially better than it regardless of what other players choose to do. (Contrast strict dominance.)

References

Dixit, A., Skeath, S., & Reiley, D. (2014). Games of strategy. New York: W. W. Norton; Company.
Hargreaves-Heap, S., & Varoufakis, Y. (2004). Game theory: A critical introduction. London: Routledge. Retrieved from http://webcat.warwick.ac.uk/record=b2587142~S1
Misyak, J. B., & Chater, N. (2014a). Virtual bargaining: A theory of social decision-making. Philosophical Transactions of the Royal Society B: Biological Sciences, 369(1655), 20130487. https://doi.org/10.1098/rstb.2013.0487
Misyak, J. B., & Chater, N. (2014b). Virtual bargaining: A theory of social decision-making. Philosophical Transactions of the Royal Society B: Biological Sciences, 369(1655), 20130487. https://doi.org/10.1098/rstb.2013.0487
Neumann, J. von, Morgenstern, O., Rubinstein, A., & Kuhn, H. W. (1953). Theory of Games and Economic Behavior. Princeton, N.J. ; Woodstock: Princeton University Press.
Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. MIT press.
Rasmusen, E. (2007). Games and information: An introduction to game theory (4th ed). Malden, MA ; Oxford: Blackwell Pub.
Tadelis, S. (2013). Game theory: An introduction. Princeton: Princeton University Press.