## Mathematics Illuminated

# Game Theory

## We've all heard it said that life is like a game. Most games have well defined rules, with clear benefits for winning and costs for losing. And that makes them something we can think about logically and mathematically. But what about life? Can mathematics tell us anything about the competitions and collaborations that happen every day? From the social sciences to biology, robotics and beyond, the answer is yes.

Competition and cooperation can be studied mathematically, an idea that first arose in the analysis of games like chess and checkers, but soon showed its relevance to economics and geopolitical strategy. This unit shows how conflict and strategies can be thought about mathematically, and in doing so, reveal important insights about human and even animal behaviors.

### Unit Goals

- Game theory is the mathematical study of social interactions as games that have payoffs for the players.
- Traditional game theory assumes that players are rational actors, always acting in ways that maximize their benefits. Real people are not necessarily like this.
- The payoff matrix is the essential way to express a game mathematically.
- In zero-sum games, a winner’s gains come at a loser’s expense.
- Non-zero-sum games can include both win-win and lose-lose situations.
- Strategies are the actions that players take in a game.
- Payoffs are often frequency dependent; that is, they depend on how many people are playing a particular strategy as well as how often a game is played.
- Equilibrium is reached when each player has no incentive to play differently.
- Game analysis changes when games are played repeatedly. This gives rise to mixed strategies.
- Game theory can provide insight into many situations, phenomena, and subjects, including biology, sociology, and linguistics.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

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Fisher, Sir Ronald Aylmer. “Randomisation and an Old Enigma of Card Play,” *Mathematical Gazette*, vol. 18 (1934).

Grassly, N.C., A. von Haeseler, and D. Krakauer. “Error, Population Structure, and the Origin of Diverse Sign Systems,”* Journal of Theoretical Biology*, vol. 206, no. 3 (2000).

Henrich, Joseph, Robert Boyd, Samuel Bowles, Colin Camerer, Ernst Fehr, Herbert Gintis, and Richard McElreath. “In Search of Homo Economicus: Behavioral Experiments in 15 Small-Scale Societies,” *The American Economic Review*, vol. 91, no. 2, Papers and Proceedings of the 113^{th} Annual Meeting of the American Economic Association (May 2001).

Henrich, Joseph. “Does Culture Matter in Economic Behavior? Ultimatum Game Bargaining Among the Machiguenga of the Peruvian Amazon,” *The American Economic Review*, vol. 90, no. 4 (September 2000).

Jager, G. “Evolutionary Game Theory and Linguistic Typology: A Case Study,” In P. Dekker, editor, Proceedings of the 14 th Amsterdam Colloquium. ILLC, University of Amsterdam, (2003).

Mero, Laszlo. [Translated by Anna C. Gosi-Greguss. English version edited by David Kramer.] *Moral Calculations: Game Theory, Logic, and Human Frailty*. New York: Copernicus Springer-Verlag New York, Inc., 1998.

Nowak M.A. and R.M. May. “Evolutionary Games and Spatial Chaos,” *Nature* 359 (1992).

Nowak M.A., S. Bonhoeffer, and R.M. May. “More Spatial Games,” *International Journal of Bifurcation and Chaos*, vol. 4, issue 1 (February 1994).

Nowak, M.A., J.B. Plotkin, and D. Krakauer. “The Evolutionary Language Game,” *Journal of Theoretical Biology*, vol. 200, Issue 2 (21 September 1999)

Nowak, M.A., Karen M. Page, and Karl Sigmund. “Fairness Versus Reason in the Ultimatum Game,” *Science*, vol. 289, issue 5485 (2000).

Poundstone, William. *Prisoner’s Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb*. New York: Doubleday, 1992.

Siegfried, Tom. *A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature*. Washington, D.C.: John Henry Press, 2006.

Von Neumann, John and Oskar Morgenstern. *Theory of Games and Economic Behavior* (6^{th }paperback edition). Princeton, NJ: Princeton University Press, 1990.