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Unformatted text preview: Ec1052: Introduction to Game Theory Handout 12.5 Harvard University 9 May 2004 Solutions to Problem Set 5 Problem 1. 1(a) Omitted. 1(b) The trust game has a unique SPE. Player 2 sends no money back and, anticipating this strategy, player 1 sends no money to player 2. Note, that player 2 essentially plays a dictator game which has a unique SPE (namely to send zero). 1(c) The amount x being sent is interpreted as trusting behavior on the part of player 1 and the share of the sent money which is sent back by player 2 measures trustworthiness. Reciprocity implies that the share y/ 3 x is increasing in x . The trust game is due to Berg, Dickhaut and McCabe: Trust, Reciprocity and Social History in Games and Economic Behavior (1995). 1(d) You could define trust as proposed above: the amount x player 1 sends in a trust game. However, any amount x is clearly not SPE - the fact that we typically observe positive amounts x being sent in lab experiments can be interpreted in two ways: The trust game is the true game being played and people simply dont play SPE in real life. Trust lies in the realm of behavioral economics just like fairness in the ultimatum game and other non-rational behavior. The trust game is not the true game. People really play repeated games with each other where a certain amount of cooperation can be sustained as SPE due to the folk theorem. If we let them play in a lab there is a halo effect: people adjust their strategies and stick to their heuristic rules even if the experimenter tells them that interaction is one-shot. Personally, I prefer the second interpretation even though the first one is very popular amongst experimental economists these days. The problem is that we might not learn much from explanation 1 even if it happens to be true. Dif- ferences in trusting behavior between populations can then only be explained by differences in those behavioral parameters. This comes awfully close to at- tributing differences in trust to culture as Fukuyama (1995) does in his book on Trust . In contrast, the second explanation allows us to correlate trust with observed variation: we can measure, for example, the frequency with which business partners interact with one another in different countries which will affect the discount factor in a repeated game. Ec1052 Handout 12.5: Solutions to Problem Set 5 2 Problem 2. The battle of the sexes game looks like this ( O for Opera and F for football). O F O 2 , 1 , F , 1 , 2 If we draw the convex hull of these payoffs we get a cone from the origin out to (1 , 2) and (2 , 1). This is the set of feasible payoffs - call it f . Let p be the probability that player 1 plays O and q be the probability that 2 does. For p 2 / 3, 2 maximizes his (stereotypically) payoff by playing O . For p 2 / 3, 2 maximizes his (stereotypically) payoff by playing F . Hence, his best response payoff as a function of p is as follows....
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