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November 29th, 2016, 09:53 AM   #1
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Cool Sequential game, how to derive utility functions.

Dear all,

I am in the process of constructing a game theoretic model related to sanctions.
We have two players, the sender (S) and the target (T). Discount factor delta.

S chooses the sanction level (w).
T chooses the action level (a).

Utility of S is increasing in (a).
Utility of T is decreasing in (a).
Utilities of S and T are decreasing in (w).

Hence, S will harm T and S by putting w(a), (w'(a)<0). Meanwhile, T chooses (a) conditional on w(a) (rational expectation).
Player-specific harm is determined by two parameters in the utility functions.

S chooses (w) at odd periods.
T chooses (a) at even periods.

I want to derive a utility function so that I can mathematically prove that the behaviour of the two players is rational. I suspect, however, that this will require dynamic programming. (I want to derive the utility function such that I can prove an equilibrium in which a(w) = w(a), i.e. the "response functions" match in equilibrium. In one sense, I want to parameterize this problem. Has anyone got a clue as to how to proceed doing this? I do not want to go by trial and error.

Please see further information below (screenshot):
https://gyazo.com/31aeb7bf005f182448f0262975c03ddd

Thank you, and have a nice day.
JF
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