What are some game theory tools and techniques to design and evaluate incentives and contracts?

1 Nash equilibrium

A Nash equilibrium is a situation where no agent can improve their payoff by unilaterally changing their strategy, given the strategies of the others. It represents a stable and self-enforcing outcome that does not require any external enforcement or coordination. You can use Nash equilibrium to analyze the optimal strategies for each agent in a game, and to predict the likely outcome of a strategic interaction. For example, you can use Nash equilibrium to design incentive schemes that induce employees to exert optimal effort, or to evaluate the effects of price competition or collusion among firms.

2 Subgame perfect equilibrium

A subgame perfect equilibrium is a refinement of Nash equilibrium that applies to games with sequential moves, where some agents act before others. It requires that the agents' strategies form a Nash equilibrium not only in the whole game, but also in every subgame, which is a subset of the game that starts at any decision node and includes all the subsequent nodes. You can use subgame perfect equilibrium to eliminate incredible threats or promises that are not credible in the subgames, and to identify credible commitments that can deter or induce certain actions by the others. For example, you can use subgame perfect equilibrium to design contracts that specify contingent payments or penalties based on observable outcomes, or to evaluate the effects of entry deterrence or reputation building by firms.

3 Bayesian games

A Bayesian game is a game where some agents have incomplete or asymmetric information about the characteristics or actions of the others, such as their preferences, payoffs, types, or signals. It requires that the agents form beliefs about the unknown information based on the available information, and update their beliefs using Bayes' rule when they receive new information. You can use Bayesian games to model situations where uncertainty and learning are important, and to derive optimal strategies that depend on the beliefs and signals. For example, you can use Bayesian games to design auctions that elicit truthful bids from buyers with different valuations, or to evaluate the effects of signaling quality or reputation by sellers.

4 Mechanism design

Mechanism design is the inverse of game theory, where you design the rules of the game to achieve a desired outcome, rather than taking the rules as given and analyzing the outcome. It involves specifying the actions, information, and payoffs of the agents, and finding a mechanism that implements the outcome as a Nash equilibrium or a dominant strategy equilibrium. You can use mechanism design to design incentives and contracts that align the interests of the agents with a social objective, such as efficiency, fairness, or revenue maximization. For example, you can use mechanism design to design markets that allocate resources efficiently, or to design contracts that elicit truthful information or optimal effort from agents with hidden information or hidden actions.

5 Cooperative games

Cooperative games are games where the agents can form coalitions and negotiate binding agreements that specify how to divide the joint payoff of the coalition. They differ from non-cooperative games, where the agents act independently and non-bindingly. You can use cooperative games to model situations where cooperation and coordination are possible and beneficial, and to analyze the stability and fairness of different coalitions and agreements. For example, you can use cooperative games to design networks that share costs and benefits among nodes, or to evaluate the effects of mergers or alliances among firms.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?