Contracts

In this article we discuss about the notion of contract, which requires understanding of Nash Equilibrium. If you are not familier with nash equiilbrium, please refer to this article.

Contract is an agreement about behavior that is intended to be enforced. Contracts are intended to be executed, but might not always be excecuted.
The following is one example of procedure that contracts can be enforced.

Players form a contract.
They play the underlying game (called production).
An external enforcer (court, arbitrator, etc. ) observes the outcome of the underlying game and compels transfers.

There are two types of enforcement : self-enforcement and external enforcement.
Self enforcement is when the nash equilibrium is already the desired result. For an example, there is the pareto coordination game.

	A	B
A	2, 2	0, 0
B	0, 0	1, 1

In this case, if players talk to each other to choose A, that contract will be executed without and external enforcement.
In contrast, external enforcement is when external entity has to compel in order to make players play the desired strategy.

Example : production game

	I	N
I	$z_1, z_2$	$y_1, x_2$
N	$x_1, y_2$	1, 1

I indicates invest, and N indicates not invest. The situation where both are investing is the desired result, but players might devicate from this strategy profile.
Assume that I, I is the most efficient outcome. So, following conditions must be true.

z_1 + z_2 > y_1 + x_2

z_1 + z_2 > x_1 + y_2

In this case, assume that individuals have incentive to deviate from intended strategy (I, I). We can make a contract here like below.

	I	N
I	$z_1, z_2$	$y_1 + \beta, x_2 - \beta$
N	$x_1 + \alpha, y_2 - \alpha$	$\gamma, -\gamma$

Note that $\alpha, \beta, \gamma \in R$ . i.e. they can also be negative numbers.
For the contract to be enforced, it should make playing I be benifical than playing N for both players.

z_1 \geq x_1 + \alpha

z_2 \geq x+2 - \beta

Trying to achieve and "efficient outcome", the payoffs must be transferable.
Verifiability constrants : The external enforcer may not be able to verify exactly what happened in the underlying game.

Full verifiability: A setting in which the external enforcer (e.g. the court) is able to perfectly differentiate between all of the different outcomes of the underlying game.
Limited verifiability: A setting in which the external enforcer is NOT able to perfectly differentiate between all of the different outcomes of the underlying game.

	I	N
I	$8, 8$	$-4, 4$
N	$10, -2$	$0, 0$

The efficient outcome is when (I, I) is played, but unfortunately (I, I) is not the nash equilibrium.
Assume there is limited verifiability.

	I	N
I	$8, 8$	$-4 + \alpha, 4 - \alpha$
N	$10 + \alpha, -2 - \alpha$	$0 + \alpha, 0 - \alpha$

$8 \geq 10 + \alpha \Rightarrow \alpha \leq -2$

$8 \geq 4 - \alpha \Rightarrow \alpha \geq -4$

There are many $\alpha$ satisfing both conditions, for example $\alpha = -3$ . But in other cases, there might be no $\alpha$ that satisfies both conditions.

The parties specify just how they want to coordinate in the underlying game. If a breach occurs, the external enforcer compels a transfer according to a legal rule. Three common ones:

Expectation damages : make the plaintiff (harmed party) as well off as he/she would have been had no breach occurred.
Reliance damages : return the plaintiff to a payoff that would have occurred with no contract (zero).
Restitution damages : return the defendant (offending party) to a payoff that would have occurred with no contract (zero).