Consider the following sequential game, where Congress and the President are try

Question

Consider the following sequential game, where Congress and the President are trying to pass a spending bill. Congress first decides to what bill to pass, at which point the President can either sign the legislation or veto it. There are two potential components of the spending bill infrastructure (I) spending which is preferred by Congress, and defense (D) spending which is preferred by the President. The legislation passed can include any combination of these components. Solve for the Subgame-Perfect Nash Equilibrium and the equilibrium path for the following game, where payoffs are in the order of (Congress, President). (3,3) Veto (2,2) Sign(4,1) (2,2) Sign(1,4) Veto (2,2) Sign (2,2) (2,2) President Si & D Veto Congress Neither Veto

Explanation / Answer

Consider the given game problem here there are two players “Congress” and the “President”. Now, “C=Congress” have four possible strategy choices and “P=President” have “two possible strategies. Now, if “C” chose “I&D”, then the optimum choice for “P” is “sign”, because “P” will get more payoff under “sign”, => under this case the payoff of both players are “3, 3”. If “C” chose “I”, then the optimum choice for “P” is “vote”, because “P” will get more payoff under “vote”, => under this case the payoff of both players are “2, 2”. So, as “C” will get more payoff in “I&D” compare to “I”, => “C” will prefer to choose “I&D”.

Now, if “C” chose “D”, then the optimum choice for “P” is “sign”, because “P” will get more payoff under “sign”, => under this case the payoff of both players are “1, 4”. So, as “C” will get more payoffs in “I&D” compare to “D”, => “C” will prefer to choose “I&D”, because “1 < 3.  

Finally, if “C” chose “Neither”, then “P” is indifferent between “sign” and “vote”, because “P” will get same payoff under “sign” and “vote”, => under this case the payoff of both players are “2, 2”. So, as “C” will get more payoffs in “I&D” compare to “Neither”, => “C” will prefer to choose “I&D”, because “3 > 2”.

So, here the solution of the game (SPNE) is “I&D, sign” and both of them will get “3” as payoff.

So, here the equilibrium path is given by, “I&D”, “Sign”.

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