Mechanism Design - A Primer

This post is an ongoing investigation of my alternative view of AI and the role of algorithms for multi-agent systems in markets.

Game theory is concerned with predicting the behavior of agents participating in an strategic interaction. We can also ask the inverse. Given a desired behavior of the agents, what strategic interaction would give rise to this behavior? For example, there are two firms $A$ and $B$. Each firm can choose to cooperate $C$ or defect $D$ which results in the following payoff matrix

The methods of game theory looks at this interaction and predicts what each agent will do. In this interaction, defecting $D$ is the strictly dominant strategy for both firms. Correspondingly, the Nash Equilibrium (NE) is $(D, D)$ with payoffs $(1, 1)$, but the socially efficient outcome is $(C, C)$ with payoffs $(3, 3)$. We can resolve this inefficency using mechanism design.

In this example, we would like to introduce a mechanism such that when the firms play the modified game, they find it in their interest to produce the socially efficient outcome. The mechanism is not changing the agents’ preferences, rather, the payoff is changed. Imagine a regulation is introduced such that a tax $\tau$ is imposed on firms that engage in $D$. The firm pays $\tau$ to the regulator regardless of what the other firms do. The payoff matrix is now

For $D$ to longer be the dominant strategy, $C$ needs to be at least as good as $D$ in both actions. In other words, the value of $\tau \geq 2$ which results in

$D$ is now weakly dominated by $C$ and $(C, C)$ becomes the NE which is the socially efficient outcome we desired. Notice that the designer of the mechanism does not force the firms to cooperate. The firms remain rational and self-interested.