Nash Equilibria

1. Nash Equilibria Fancy graphs make everything look more official! By Kallen Schwark 6/11/13

2. Nash what? Is that a new tissue brand? • A Nash equilibrium is “a set of strategies, one for each player, such that no player has incentive to unilaterally change his or her action.” • Basically, everyone acted optimally for themselves when considering the actions of everyone else. • If any ONE PERSON changed his or her own strategy, that person’s payoff would decrease or remain the same, NOT increase. • (Note: some games involve strategies that depend on probability for their payoffs. Nash equilibria here are those that will result in a decrease in EXPECTED payoff if a player makes a unilateral change in strategy.)

3. Subgame Perfection (no, not Battleship)(get it) (SUBgame perfection) (like submarines) • Nash equilibria sometimes only work because certain situations in the game are avoided; if the game were started in these situations, the strategies used by the players would no longer form a Nash equilibrium. • These strategies use non-credible threats, which are parts of one player’s strategy that would decrease all players’ payoff should the other player take a certain action; if the other player acted in this way, the first player would choose a different strategy out of self-interest. “If you take my candy, I will set off this bomb in my hand!” • In contrast, some remain Nash equilibria in all possible parts of the game; this is a state of subgame perfection.

4. John Nash: Judging by his name, he sounds important. • Nash showed that games with finite sets of actions always have an equilibrium involving mixed strategies, or strategies using probability to determine the action. • Suppose there is a game in which players A and B flip coins. If the coins produce the same result, A wins $10. If different results, B wins $10. The mixed-strategy equilibrium here is that both players use a strategy that picks heads 50% of the time; by randomizing the actions used, the highest EXPECTED payoff is achieved by both players. • Nash also theorized that predicting the result of choices of multiple decision-makers by analyzing them in isolation is impossible. (He was kind of important to the field of game theory, so they named the equilibrium after him.)

5. Nash Equilibria and Pareto optimality (it’s the best possible outcome – but not really) • Pareto optimality is a fancy-schmancy way of stating whether it is possible to increase the payoff of one player without decreasing the payoff of another. • If this is possible, the result (and set of decisions) is NOT Pareto optimal. • Nash equilibria are NOT ALWAYS Pareto optimal. Consider the Prisoner’s Dilemma, whose normal form table is shown:

6. Nash Equilibria and Pareto optimality (cont.) • The PD Nash equilibrium is that both players tell (the top-left corner in the table), as defecting will always provide a higher payoff than cooperating in this case. • However, it is clearly not Pareto optimal; it is possible to improve the payoff of both players by having them cooperate (bottom-right corner) • This situation seems to prove that, even though there is no motivation for individual players to deviate from the equilibrium, it is in one’s true self-interest to NOT be a traitorous scumbag. (Did you hear that, Benedict Arnold?)

7. Applications to Social Choices(Wait, we’re learning things that matter?) • Nash equilibria represent steady states, or situations in which no one involved has any motivation to deviate from the norm, as the equilibrium provides them with the highest payoff when other players’ strategies are considered. • This causes stability and is extremely favorable in social situations, provided that everyone knows what everyone else is doing. • Nash equilibria can: • Predict the actions of the general public • Suggest possible solutions to unfavorable or harmful equilibria • Torture mathematics of social choice students such as Emily • Help construct accurate models of human behavior

8. Applications to Economics(MONEY!!!!!11 $$$$$$$) • If companies, consumers, or other economic entities are swapped in for the roles of players, Nash equilibria can represent an economic model to represent interactions between different companies (cooperation vs. competition) and resulting supply and demand. • This would help economists understand and predict the logical price fluctuations in certain industries. • Also, it lends to a better understanding of the economy, which aids in becoming rich. Just an added thought. ~*~$~*~

9. Bibliography David, Albouy. "Nash Equilibrium, Pareto Optimality and ." University of California at Berkeley. Berkeley. 2008. Web. 12 June 2013. Osborne, Martin J. An Introduction to Game Theory. N.p.: Oxford University Press, 2002. 11-52. University of Toronto. Web. 12 June 2013. Sethi, Rajiv. "Nash Equilibrium." International Encyclopedia of the Social Sciences. 2nd ed. 2008. 540-42. Web. Shor, Mikhael. "Nash equilibrium." Game Theory.net. N.p., n.d. Web. 11 June 2013. <http://www.gametheory.net/dictionary/NashEquilibrium.html>.