Introduction to Experiments on Game Theory

Transcription

1 Introduction to Experiments on Game Theory Syngjoo Choi Spring 2010 Experimental Economics (ECON3020) Game theory 1 Spring / 23

2 Game Theory A game is a mathematical notion of a strategic interaction in which players payo s depend on their own and others decisions. sports games such as soccer; bargaining between a rm and a labor union; bidding for things on ebay, etc. Game theory is a mathematical tool used to analyze such strategic interactions and predict people s behavior. A game is characterized by the (number of) players; their sets of feasible actions; the information available at each decision point; the payo s as functions of all decisions and random events. Experimental Economics (ECON3020) Game theory 1 Spring / 23

3 Game Theory and Experimental Method A usual question: the theory is interesting... but do people actually play this way? Behaivor in games is notoriously sensitive to details of the environment. To test game theory, we can use data that naturally occur in eld settings (such as auctions). But the use of eld data is often limited by missing variables on details of strategic environments. In contrast, experimental control provides a decisive advantage in identifying the relationship between behavior and environment. Experimental Economics (ECON3020) Game theory 1 Spring / 23

4 Some Basics in Game Theory I On one hand, games can be divided by the structure of movement: In static (or simultaneous-move) games, players simultaneously choose actions (or do not know others choices when each of them makes a decision); In dynamic games, players choose actions in some sort of order and observe the history of the play of the game. On the other hand, games can be divided by the structure of information: In games of complete information, each player s payo function is common knowledge among all the players; In games of incomplete information (called Bayesian games), at least one player is uncertain about another player s payo function. Experimental Economics (ECON3020) Game theory 1 Spring / 23

5 Some Basics in Game Theory II Static games of complete information with 2 players and nite actions for each player are often represented by a matrix form: A B A u 1 (A, A), u 1 (A, B), u 2 (A, A) u 2 (A, B) B u 1 (B, A), u 1 (B, B), u 2 (B, A) u 2 (B, B) Each player has two action choices, A or B. u i (A, B) represents player i s utility (or payo s) when player 1 chooses A and player 2 chooses B. For other pairs of actions, the interpretation is the same. Experimental Economics (ECON3020) Game theory 1 Spring / 23

6 Some Basics in Game Theory III A key notion of a game involves a strategy, which is essentially a complete plan of action that covers all contingencies. In the previous matrix game, each player s strategy set is fa, Bg; In either games of incomplete information or dynamic games, a strategy speci es what a player will do in each contingent situation. In order to predict which strategies are chosen by players, economists rely on an equilibrium that speci es a set of strategies in which each single player has no desire to deviate and choose any other strategy. John Nash (1950) provided a notion of equilibrium, called Nash equilibrium, in a general environment of strategic interaction. Experimental Economics (ECON3020) Game theory 1 Spring / 23

7 Dominance Strategy A strictly (weakly) dominates strategy B if the payo from choosing A is higher than (at least as high as) the payo from B, for any strategy choice by other players. A B A $10, $18, $14 $6 B $4, $7, $20 $8 This notion of dominance is extremely appealing because A will turn out better than B no matter what you think other players will do. Thus, it is natural that one choose a dominant strategy even if one does not know how rational other players are. Experimental Economics (ECON3020) Game theory 1 Spring / 23

8 Let s think about how you will play the following game: A B A $80, $0, $80 $100 B $100, $35, $0 $35 This is a so-called Prisoner s Dilemma game. The dominance argument predicts that both players choose B, which results in $ 35 for each player. In fact, this is a prediction by Nash equilibrium. Experimental Economics (ECON3020) Game theory 1 Spring / 23

9 Here the cooperative player is strategy A. What s going wrong against the equilibrium? (One possible story) Cooperative play (A) can increase the total welfare of both players. Players might not be just sel sh. Experimental Economics (ECON3020) Game theory 1 Spring / 23 A Prisoner s Dilemma Experiment Cooper, DeJong, Forsythe, and Ross (1996, GEB) ran an experiment in which individual subjects were prevented from being rematched with the same person:

10 A Classroom Experiment Now let s play a game in a classroom experiment. Go to Session name is. Experimental Economics (ECON3020) Game theory 1 Spring / 23

11 A Classroom Experiment The game you just participated in is called a guessing game or a beauty-contest game, which was rst experimentally studied by Nagel (1995). A key aspect of most games is the need for players to guess what others will do in order to determine their own best decision. This aspect is somewhat obscured in a 2 2 game since a wide range of beliefs may lead to the same decision. Nagel s guessing game with a continuum of decisions, from 0 to 100, may reveal what guess each player make about the average. Experimental Economics (ECON3020) Game theory 1 Spring / 23

12 Iterated Dominance Dominance can be applied iteratively. rst eliminate dominated strategies for all players; then check whether that rst round of elimination makes some (initially undominated) strategies dominated eliminate those (iteratedly) dominated strategies, and repeat this process as long as possible. Games in which this process of iteratively deleting dominated strategies leads to a unique equilibrium are called dominance solvable. In order to do this iterated process, one must believe that others obey dominance as well but it may be less obvious in practice (!). Experimental Economics (ECON3020) Game theory 1 Spring / 23

13 John M. Keynes Quotation - Beauty Contest In his famous book, General Theory of Employment, Interest, and Money, he draws an analogy between the stock market and a newspaper contest in which people guess what others will guess are most beautiful (1936, p.156): "It is not the case of choosing those which, to the best of one s judgement, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree, where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fth, and higher degrees." Experimental Economics (ECON3020) Game theory 1 Spring / 23

14 Guessing Games In the guessing games you participated in, each of N players, i, chooses a number x i in the interval [0, 100] simultaneously; the target numberis 0.5 of the average of all chosen numbers, 0.5 N i =1 x i /N (in the second game, it is N i =1 x i /N ); The player whose number is closest to the target number wins a xed prize. The Nash equilibrium (NE) in the rst game is that all choose 0. In the second game the NE is that all choose 40. Why? Experimental Economics (ECON3020) Game theory 1 Spring / 23

15 NE in Guessing Games In the rst game, note that choosing any number in the interval (50, 100] is weakly dominated by 50; In the rst step of iterated dominance, each player can eliminate (50, 100]. Suppose now each player believes that all others obey dominance and thus the strategy set amounts to [0, 50]. In the second step of iterated dominance, each player can eliminate (25, 50] since any number in this interval is dominated by 25. And so on... In nitely many steps of iterated dominance lead to the unique NE, which is that all players choose 0. In the second game, the NE is that all players choose 40. (just check whether any individual has an incentive to deviate from 40, when all others choose 40.) Experimental Economics (ECON3020) Game theory 1 Spring / 23

16 Experimental Results I - Initial Play Nagel (1995) ran experiments according to the rst treatment in our classroom game. The results for the rst-period play in her experiments is summarized: Experimental Economics (ECON3020) Game theory 1 Spring / 23

17 Experimental Results II - Repetition In Nagel s experiments subjects replicated the same game four times and the players chose lower numbers but not 0 at later periods. Experimental Economics (ECON3020) Game theory 1 Spring / 23

18 Experimental Results III - Newspaper Experiments Bosch-Domènech, Montalvo, Nagel, and Satorra (American Economic Review, 2002) ran the same experiments using the advertisement in three Newspapers: Financial Times (UK), Expansión (Spain), and Spektrum (German). The target number is (2/3) N i=1 x i /N. Experimental Economics (ECON3020) Game theory 1 Spring / 23

19 Comments from Participants in the Newspaper Experiments Experimental Economics (ECON3020) Game theory 1 Spring / 23

20 Experimental Economics (ECON3020) Game theory 1 Spring / 23

21 S#1206: In case that all numbers are Experimental Economics (ECON3020) Game theory 1 Spring / 23

22 Experimental Economics (ECON3020) Game theory 1 Spring / 23

23 What Can We Learn From These Experiments? Sometimes, people s behavior is not entirely driven by self interest. Other motivations like altruism might be important in economic environments. We will pursue this issue more in Week 4. Some people think iteratively about what others will do, but this is not uniform, and the e ect of such introspection is not enough to move decisions to near-nash levels in a single round of play. When people are able to learn from experience, behavior does converge to Nash equilibrium in this game. We will pursue the last two issues further in Week 5 and 6. Experimental Economics (ECON3020) Game theory 1 Spring / 23