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A major lesson from game theory seems to be that in simultaneous move games, a player does not change mixing proportions in response to changes in their own payoffs. Rather, their opponents change mixing proportions instead, while the player changes mixing proportions only in response to changes in their opponents payoffs.

How common is it for this behavior to actually occur? Is it mostly a toy model that only applies to very controlled decisions such as whether to cheat to the left or right in sports, or is it observed more ubiquitously?

I have difficulty believing that changing the penalty for lying about taxes only affects the proportion of people who get audited, and does not affect the proportion of people who lie about taxes in the real world, even though I understand the math involved in finding the Nash Equilibrium. It is also difficult to believe that the equilibrium effect of hiring more police is that punishment severity will go down, while the equilibrium effect of reducing punishment severity is that more police will get hired, with neither having any impact whatsoever on crime rate. This type of causal pattern certainly seems to contradict the usual discourse about incentive structure. How seriously should one take this conclusion into account when predicting the effects of various incentive structure changes in everyday life?

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    $\begingroup$ Most people are not psychopaths who only seek to maximize monetary payoff in every situation. If I knew I would not be caught stealing candy from the store, I would still not do it. $\endgroup$
    – Giskard
    Aug 31, 2022 at 5:14
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    $\begingroup$ Also, there is little evidence that capital punishment deters crime, so the straightforward application does not seem to work there either. $\endgroup$
    – Giskard
    Aug 31, 2022 at 5:19

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I don't think the major lesson is quite that. If an expected payoff maximizing player mixes between several pure strategies, they must be indifferent between playing all of them. This has the curious effect that in a completely mixed Nash equilibrium, a player randomizes in exactly the way that makes the other player indifferent.

But there is no reason to restrict oneself to completely mixed Nash equilibria. Some people lie about taxes, some don't. For most of these people, only one choice might be optimal. Harsher punishments might mean that agents move from the former category to the latter.

Nevertheless, in lab experiments, subjects seem to behave in ways that contradict the standard argument even for games in which equilibria mus be completely mixed. See Example B in

Goeree, Jacob K., and Charles A. Holt. "Ten little treasures of game theory and ten intuitive contradictions." American Economic Review 91.5 (2001): 1402-1422.

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