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Nagel: Unraveling in Guessing Games: An Experimental Study Economics 328 Spring 2005

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What is Game Theory? Game theory is the study of strategic interaction among individuals or groups. I have to anticipate what others will do in order to determine my own best strategy. I should anticipate that others will be trying to figure out what Im going to do. Examples of Games Many industries are dominated by a small number of large firms. In deciding on a competitive strategy, such firms cannot take the market as fixed. Instead, firms should anticipate other firms responding to their actions. Rather than being driven by ideology, many politicians are quite strategic in choosing their positions. In choosing a position, politicians must anticipate not only how the electorate will react but also how opposing politicians will react. Many situations from everyday life have strong strategic elements. For example, think about choosing where to go with your S.O. on the weekend. What are strategic elements of this situation?

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What is Game Theory? Game theorists try to develop simple models of strategic situations and then predict how individuals will behave in these games. These models almost always rely strongly on the rationality of players. Experiments have played two important roles in the development of game theory. Experiments have unequivocally demonstrated that most individuals are not as rational as game theorists might like. Experiments have helped identify systematic departures from rationality. By building these systematic departures into the theory, we can hopefully build a theory with greater predictive power.

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Normal Form Games Definition: A normal form game consists of the following elements: A set N of players. Let n be the number of players in N. For each player i, a set S i of available strategies. A strategy summarizes every action a player will take under any conceivable situation in the game. For each player i, a payoff function u i (s 1,s 2,…,s n ). This function gives player i's payoffs as a function of his/her strategy and the strategy choices of all n-1 other players. Often times, we represent normal form games in tabular form.

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Normal Form Games -- Example

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Dominance One of the simplest solution concepts for games is dominance. Individuals should never use a strategy if the exists another option that always does (weakly) better. This requires only a low level of rationality. Definition: A strategy s i is (weakly) dominated for player i if he has another strategy s i such that s i never gives a (weakly) greater payoff than s i regardless of what strategies are chosen by the other players. Example: In the arbitration game, suppose.02W c L.21W. For both sides it is a dominant strategy to use a lawyer. This is an example of a prisoners dilemma – both sides would be better off if they could coordinate on not using lawyers.

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Iterated Removal of Dominated Strategies The set of serially undominated strategies is defined iteratively. Start with some normal from game {N,S,u}. Form a new game by removing all of the weakly dominated strategies from the old game. Take the new game and form a third game by removing all of the weakly dominated strategies from the new game. This process continues until we cant remove any more strategies. The remaining strategies are called the set of serially undominated strategies. The process of removing strategies is called iterated removal of weakly dominated strategies. Like dominance, iterated removal of dominated strategies requires rationality. Moreover, common knowledge of rationality is also required. I am rational. I know you are rational. I know you know I am rational I know you know I know you are rational. And so forth, ad infinitum

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The Beauty Contest Game Consider Nagels guessing game, also known as the beauty contest game. The strategy set available to each player in this game is the numbers between 0 and 100 (inclusive). The player closest to (for example) ½ the median number chosen wins a prize of $1. All other individuals win nothing. In case of a tie, the prize is split evenly. We can show that zero is the only serially undominated strategy. Why is it the beauty contest game?

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The Beauty Contest Game Round 1: The median can never be greater than Therefore, picking 500 weakly dominates any higher number. Remove the numbers greater than 500. Round 2: Given that only the numbers from 0 to 500 remain, the median can never be greater than 500. Therefore, picking 250 weakly dominates any higher number. Remove the numbers greater than 250. Round 3: Given that only the numbers from 0 to 250 remain, the median can never be greater than 250. Therefore, picking 125 weakly dominates any higher number. Remove the numbers greater than 125. We can continue this process of elimination ad infinitum. The only number that won't eventually be eliminated is zero.

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Nagel: Unraveling in Guessing Games Research Questions: The guessing game is good setting for seeing how subjects reason about games. The main question is whether subjects will behave as rationally as the theory predicts. If not, can we see subjects learning over time? In particular, will they get better at anticipating the actions of others? Initial Hypotheses: Rather than being fully rational, Nagel expected subjects to only go a few steps into the chain of reasoning. More technically, she expected to see play in the first game clustered around 100*pn, where n is some integer, and then clustered around R*pn in later rounds, where R is a reference point. She expected subjects to learn over time, possibly even going more steps into the chain of reasoning. Experimental Design: Nagel ran the guessing game with subjects guessing between 0 and 100. She ran sessions with p = 1/2 (3 sessions), p = 2/3 (4 sessions, and p = 4/3 (3 sessions). Note that iterated dominance no longer applies with p = 4/3. Sessions contained 15 – 18 subjects. Each session played the game four times with standard feedback. Sessions were fairly short (about 45 minutes) and the prize was substantial.

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Experimental Results Substantial clustering is observed, as predicted by a steps of reasoning approach. In the first game, most of the clustering takes place at steps 0, 1, and 2. Choosing step 2 is in fact typically optimal given the choices of the population – playing the equilibrium would actually lead to zero earning. Notice that with p = 4/3, subjects go fewer steps into the reasoning – one step is the modal choice rather than two steps (as for p = ½ and p = 2/3).

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Experimental Results Subjects clearly move towards lower choices over time in the p = 1/2 and p = 2/3 treatments. They move toward higher choices over time in the p = 4/3 treatment.

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Experimental Results

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Clustering is not as clear in later round, possibly because it is less easy to identify the reference point. There is little evidence of increased depth of reasoning with experience. Nagel claims that her data support the directional learning hypothesis. My opinion is that this hypothesis is poorly developed and that her results are indistinguishable from regression to the mean.

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Follow-up Papers Camerer, Ho, and Weigelt (1998) revisit the guessing game (renamed the p- beauty contest game). They add several details to Nagels findings: They use a more rigorous method than Nagel to estimate levels of reasoning, and find somewhat lower levels of reasoning (more level 0s, fewer level 2s and 3s). They study the ability of subjects who are experienced with one p-beauty contest game to play a different p-beauty contest game. They find some evidence of positive transfer – initial behavior is the same for experienced and inexperienced subjects, but experienced subjects learn faster. They study the learning process more carefully than Nagel. They find evidence that a large fraction of the population (about 70%) is able to anticipate learning by others. Bosch-Domènech, Montalvo, Nagel, and Satorra (2002) report results from a series of newspaper experiments with the guessing game (p = 2/3). In spite of the lack of control, the newspaper experiments look remarkably similar to the laboratory experiments with spikes at 33, 22, and 0. Subjects who were able to correctly describe the Nash equilibrium generally (81%) did not choose zero. Subjects who conducted their own experiments did better than game theorists or the general public at predicting the winning number!

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