Overview Game theory is a branch of logic which deals with cooperation and conflict in the context of negotiations and payoffs. The theory of games can elucidate the incentive conditions required for cooperation, can aid understanding of strategic decisions of nations or actors in conflict, and can help in the development of models of bargaining and deterrence.

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography





Key Concepts and Terms

• The Prisoner's Dilemma is the classic game in game theory literature. It centers on a game in which both actors would be better off cooperating, but both have an individual incentive to defect (not to cooperate) and as a result the likely outcome is one which is worse for both players than had they cooperated.

• Repeated games. In real life, most games are repeated rather than single-shot. Repetition means each player has additional information based on past game decisions of the other player. This complicates calculation of choices and changes the equilibrium point. See Fink, Gates, and Humes, 1998: ch. 3. For instance, if the Prisoner's Dilemma is repeated a sufficient number of times, for instance, players may learn to take a strategic view and cooperate.

• Strategies. A strategy is a plan of action that cannot be upset by an opponent or nature. The purpose of strategies is to secure the most favorable game value in the long run. A common strategy is tit-for-tat, in which the player responds to a given game move with a mirroring move. A pure strategy involves always pursuing the same strategy. A mixed strategy involves randomly choosing among one's best strategies according to some proportions in order to maximize favorable game value when a pure strategy in repeated games would give the opponent an advantage of predictability. The part of a proportion (such as the 3 in the proportion 1:3) corresponding to a particular strategy is called the oddment of that strategy.

• Game matrix. A game matrix is the table of all strategies of person A (as columns) versis all strategies of person B (as rows). The cell entries in a game matrix are payoff values
  Maximin, minimax, and saddle point. Consider the following tables for zero-sum games:

  TABLE I Person A     Strategy 1 Strategy 2 Person B Strategy 1 -1 -25   Strategy 2 0 -20

  TABLE II Person A     Strategy 1 Strategy 2 Strategy 3 Person B Strategy 1 7 6 4   Strategy 2 3 2 5

  
  
  In the tables above, the payoffs are to Person B, following convention. Thus, in Table I, when Person A pursues strategy 2 and Person B pursues strategy 1, the payoff to B is -25 units. Using conservative reasoning, Person B will select the strategy where the least to be gained is highest (or the most to be lost is lowest). The row containing the highest minimum (the maximum row minimum, or maximin) for Person B in Table I is the row for Strategy #2, where the maximin is -20. Person A will seek the column containing the lowest maximum (the minimum column maximum, or minimax), where the minimax is the same cell,-20. When the same cell is both the maximin and the minix,it is the saddle point. Any saddle point is also the solution to the game because it will the the payoff which results when the game is played by opponents using conservative rationality.
  • Reduced games. Table II above represents a more complex game in which Person A will always prefer strategy 2 over strategy 1. That is, for Person A, strategy 2 dominates strategy 1. Recall the payoffs shown in Table II are to B. If Person B chooses strategy 1, Person A will lose only 6 points rather than the 7 if A chooses strategy 1. Likewise A will only 2 rather than 3 points if A selects strategy 2 and B also selects strategy 2. Since Column 2 dominates column 1, column 1 may be dropped, resulting in the reduced game in Table III:

    TABLE III
    Person A
    		Strategy 2	Strategy 3	B's Odds
    Person B	Strategy 1	6	4	3
    	Strategy 2	2	5	2
    	A's Odds	1	4

  • Odds in games without saddle points. In the reduced game in Table III, there is no saddle point. The maximin is 4 but the minimax is 5. Person B will prefer strategy 1 because 4 is the least to be won (and possibly 6), whereas with strategy 2 the least to be won is 2 (or possibly 5). Person A will prefer strategy 3 because the most to be lost is 5, whereas under strategy 2 the most to be lost is 6. However, person A may want to play strategy 2 once in a while if Person A is consistently playing his strategy 1 because then Person B will lose only 2, Person A can't do this consistently, however, because then Person B will move to strategy 2 and Person A will lost 5. Person A wants to usually play strategy 3, but "sneak in" a strategy 2 once in a while. The proportion for the "once in a while" is determined by the odds. To compute the odds for Table III, subtract the cells within each column or row, ignore minus signs, and place the answer in the opposite column. Thus, for instance, 6 -1 =4 and 4 - 5 =-1, giving 1 and 4 for A's odds, which means A should randomly use strategy 3 once for every four times strategy 3 is used.

  • Fair game value. The value of a game equals either person's odds played against any single strategy of the opponent. Thus, for Table III above, [(3x6)+(2x2)]/(3+2) = 4.4. Since a fair game has a value of 0, B should make a side payment of 4.4 units to A before each game if the game is to be fair.

  • Other types of games. Note that there are many other types of games than zero-sum games played under conservative rationality. Assumptions about rationality may be varied, for instance, and games may be cooperative rather than competitively zero-sum. Also, payoffs may be ordinal rather than interval and information may not be full.

  
  

  Assumptions

  • Game theory usually assumes players respond rationally based on payoffs in the game. The most common assumption is one of conservative rationality, in which players choose strategies that assume maximum average gains or minimum average losses.

  • Most applications of game theory assume conditions of full information, a condition rarely met in the real world.

  
  

  Frequently Asked Questions

  • Is game theory purely a branch of logic?
    No, political scientists have long been interested in a behavioral, approach to game theory, testing out the implications of formal game theory using small group experiments.

  
  

  Bibliography

  • Binmore, K. (1992). Fun and games: A text on game theory. Lexington, MA: D. C. Heath.
  • Dresher, M. (1961). Games of strategy: Theory and applications. Englewood Cliffs, NJ: Prentice Hall, 1961.
  • Fink, Evelyn C., Scott Gates, and Brian D. Humes (1998). Game theory topics: Incomplete information, repeated games, and n-player games. Quantitative Applications in the Social Sciences Series No. 122. Thousand Oaks, CA: Sage Publications.
  • Gates, S. and B. D. Humes (1997). Games, information, and politics: Applying game theoretic models to political science. Ann Arbor, MI: University of Michigan Press.
  • Hargreaves Heap, S. and Y. Varoufakis (1995). Game theory: A critical introduction. London: Routledge.
  • Luce, R. D. and H. Raiffa (1957). Games and decisions: Introduction and critical survey. NY: Wiley and Sons.
  • Maynard Smith, J. (1982). Evolution and the theory of games. Cambridge, UK: Cambridge University Press.
  • Morrow, J. (1994). Game theory for political scientists. Princeton, NJ: Princeton University Press.
  • Myerson, R. (1991). Game theory: Analysis of conflict. Cambridge, MA: Harvard University Press.
  • Ordeshook, P. C. (1986). Game theory and political theory. Cambridge, UK: Cambridge University Press.
  • Rapoport, A. (1966). Two-person game theory. Ann Arbor, MI: University of Michigan Press.
  • Rasmusen, E. (1989). Games and information. Cambridge, UK: Basil Blackwell.
  • Riker, William (1967). Bargaining in a three-person game. American Political Science Review, 61(3): 642-656.
  • Smoker, P. (1973). International relations simulations: A summary. In H. Alker, K. Deutsch, and A. Stoetzel, eds. (1973). Mathematical approaches to politics. San Francisco, CA: Jossey Bass: 417-464.
  • Von Neumann, J. and O. Morgenstern (1944). Theory of games and economic behavior. Princeton, NJ: Princeton University Press. The seminal work in game theory.
  • Zagare, F. (1984). Game theory: Concepts and applications. Quantitative Applications in the Social Sciences Series No. 41. Thousand Oaks, CA: Sage Publications.

Copyright 1998, 2008 by G. David Garson.
Last update: 3/25/08.