Game Theory | Philosophical Psychology

Wouldn’t it be cool if we had a way of saying, for any situation, how interacting rational agents would act? Suppose we could specify a general recipe which would tell us, for any situation at all, which actions rational agents would perform. Wouldn’t that be useful to understanding social interactions?

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‘we wish to find the mathematically complete principles which define “rational behavior” for the participants in a social economy, and to derive from them the general characteristics of that behavior’

\citep[p.~31]{neumann:1953_theory}

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Here is a very simple situation in which you face a choice. Assuming you prefer £10 to £10, I can predict which box you will open.

Alternatively, observing which box you open will tell me whether you prefer to have £10 or £0. (significance [for later]: Revealed Preference interpretation)

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But now consider this situation. Here it is not just your choice that determines the outcome, but also mine. But you don’t have an information about what I will do. So now there’s nothing for you to do but pick a box at random.

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The tacit assumption is that I am getting nothing no matter what I do. But suppose we change the game slightly ... suppose you offer me £2 to put the money in box A. Then your situation changes ...

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If I prefer £2 to £0[, and if I am rational, and ...] then I will put the money in box A. If you know all this, you can predict my action. And if you can predict my action, you can rationally choose to open box A.

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Let me pause over this. Suppose that I don’t care about your reward, only my own. Suppose also that I prefer £2 to £0. Then regardless of what you do, I should put the money into box A

This is a relatively simple interaction: the outcome my actions bring about for me does not depend at all on what you do.

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By contrast, which outcomes your actions bring about depends on what actions I select.

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Note also that it is rational for you to choose box A even if your preferred outcome would be to get £10 rather than £8. As a rational agent, you want to best satisfy your preferences. But of course you can’t just follow the money: instead you have to take into account how I am likely to act.

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Consider this profile of actions ... ... you might think that these are the most rational actions to perform since they give each Prisoner what she most prefers. But note that:

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Prisoner X can improve the ouctome by unilaterally deviating from this profile ...

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So this is the only nash equilibrium.

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\citep[p.~31]{neumann:1953_theory}

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A nash equilibrium for a game
is a profile of actions
from which no agent can unilaterally profitably deviate

Let’s see another example

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Consider this profile of actions ... ... you might think that these are the most rational actions to perform since they give each Prisoner what she most prefers. But note that:

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Prisoner X can improve the ouctome by unilaterally deviating from this profile ...

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So this is the only nash equilibrium.

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Why is the notion of a nash equilibrium so cool? Consider:

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Consider all this complexity. The notion of a nash equilibrium cuts it out. It allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed. Or does it?

		me
		put £10 in box A	put £10 in box B
you	open box A	£10 £0	£0 £0
	open box B	£0 £0	£10 £0

		me
		put £10 in box A	put £10 in box B
you	open box A	£8 £2	£0 £0
	open box B	-£2 £2	£10 £0

		Prisoner X
		resist	confess
Prisoner Y	resist	3 3	0 4
	confess	4 0	1 1

		Prisoner X
		resist	confess
Prisoner Y	resist	3 3	0 4
	confess	4 0	1 1

		Gangster X
		back off	fight
Gangster Y	back off	3 3	1 4
	fight	4 1	0 0

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