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ECON 3801, Implementation theory and Hold-up problem

Professor is away, and the TA teaches today!

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Implementation Theory

The payoff of a game is dependent on the state, which make a specific strategy not always optimal. The players knows which state they are playing.

Consider the Hawk-Dove game with 2 states, blue and red. Red is essentially the flip of Blue, and the occurance of both scenario is 1/2. The expected payoff is 11, but a well-coordinated game can increase the payoff to 14, for example a contract under different states. Can each player benefit from deviating? Yes. Can they reach the social optimal? No.

red	hawk	dove
hawk	7,7	3,8
dove	8,3	2,2

blue	hawk	dove
hawk	2,2	8,3
dove	3,8	7,7

But how to achieve the social optimal? Imagine there is a judge that can punish the person who deviates from the agreement:

red	hawk	dove
hawk	7,7	3+2,8-2
dove	8-2,3+2	2,2

blue	hawk	dove
hawk	2,2	8-2,3+2
dove	3+2,8-2	7,7

Thus, the equilibrium now is (h,h) in the red scenario and (d,d) in the blue scenario.

However, the judge cannot tell which scenario they are in. Judge cannot add or reduce the payoff based on the state, but only always exerts an addition or deduction, which, in reverse, incentives the deviation.

The state is not verifiable… why not we punish both of them?

The judge does not have to figure out who is the good guy who is the bad guy, but punishing all people once deviation is observed:

red	hawk	dove
hawk	7,7	3-2,8-2
dove	8-2,3-2	2,2

blue	hawk	dove
hawk	2,2	8-2,3-2
dove	3-2,8-2	7,7

and we achieved the social optimal. Yay!

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Hold-up Game

The game is set up as:

• One Buyer and one Seller (Bishop and Soprano)
• Customized practicing cost i (sunk and fixed cost)
• Performance cost (fixed) C < V1, which implies that the performance is always efficient
• value of the performance V1 and V2, which is based on whether the Soprano practices beforehand

The time sequence of the Game is:

1. Contract Design
2. Decide practice or not
3. Perform or not

Best outcome: practice and perform, and the social values remains the same. What we need to do is to distribute the value between the buyer and the seller (enjoyment from the performance and paycheck from performing). Assume both of them have equal bargaining power, the surplus should be divided half-half. This avoids the situation that the Soprano gains revenue by not practicing while the bishop gains nothing.

Similar to the cake splitting game, the nash equilibrium of social optimal is achieved under the shared assumption that: “each penny i get is get hold-up by the buyer”.

Any other implementation approach?

Let’s write the game in the table form:

blue	I practice	i didn’t
you practice	0, v2 - i -c	judge
you didnt	judge	v2-v1, v1-i-c

And the judge intervenes when they have a disagreement, even not knowing the actual situation. “I’ve got a double-barrel shotgun and if you come to the court, I will shoot both of you without knowing who is guilty.”

In the real world, the contract is the weapon that regulated the behavior.

Subgame Perfect Implementation Approach

We know the equilibrium, then we need to design a game to achieve the equilibrium!

Consider the game as an extensive-form game, that Soprano moves first… This divides the scenario into 4 forms, each consist of whether the soprano claims and actually practice or not.

and i lose track of the lecture… 注意力涣散