Question Session 07

These are the slides I prepared for the question session. In the end we had a small-group discussion about just part of this. Because the event turned into a discussion, there is no recording. You are of course welcome to ask questions.

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Notes

Mixed Strategies (Alex’ Question)

Optional. You do not need to know about mixed strategies for this course.

In many situations you might want to vary how you act rather than always acting in the same way. You may benefit from making your actions unpredictable to others. The game-theoretic notion of a mixed strategy is intended to capture this.

To illustrate, suppose you are playing the game rock, paper, scissors (see Index of Games). It would not be good if your opponent could predict your action. How might you play? One possibility would be to try to pick each action with equal probability, but in a way that was unpredictable. To illustrate, you might toss a three-sided dice and decide what to do based on which side it lands on.

In a mixed strategy, one or more players does not simply select an action to perform but rather assigns weights to the different actions and then selects one at random in such a way that the probability of selecting an action matches the weight assigned to it.

To illustrate, in hawk-dove (see Index of Games), Gangster Y might decide to play stay home with probability 0.75 and attack with probability 0.25.

They expected payoff from a mixed strategy is obtained by calculating the expected payoff for each action and multiplying it by the probability that the action will be performed if the mixed strategy is implemented. (See Tadelis (2013, p. §6.1.4) for details.)

The notion of a Nash equilibrium can be extended to mixed strategies:

‘Nash equilibrium is defined as a list of mixed strategies, one for each player, such that the choice of each is her best choice, in the sense of yielding the highest expected payoff for her, given the mixed strategies of the others.’ (Dixit, Skeath, & Reiley, 2014, p. 216; see Osborne & Rubinstein, 1994, p. definition §32.3 for a more formal statement)

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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, Morgenstern, Rubinstein, & Kuhn (1953) and presented in any of the standard textbooks including. Hargreaves-Heap & Varoufakis (2004); Osborne & Rubinstein (1994); Tadelis (2013); Rasmusen (2007).

mixed strategy : In game theory, a mixed strategy for a player is a probability distribution over the actions available to the player.

Nash equilibrium : a profile of actions (sometimes called a ‘strategy’) from which no agent can unilaterally profitably deviate.

References

Chater, N. (2018). The Mind is Flat: The Illusion of Mental Depth and The Improvised Mind. Penguin UK.

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

Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–292.

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.

Simonson, I. (2015). Mission (Largely) Accomplished: What’s Next for Consumer BDT-JDM Researchers? Journal of Marketing Behavior, 1(1), 9–35. https://doi.org/10.1561/107.00000001

Tadelis, S. (2013). Game theory: An introduction. Princeton: Princeton University Press.

Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211(4481), 453–458. https://doi.org/10.1126/science.7455683

Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323. https://doi.org/10.1007/BF00122574