How do you manage conflicting objectives in game theory?

1 Nash equilibrium

A Nash equilibrium is a situation in which no player can improve their outcome by unilaterally changing their strategy, given the strategies of the other players. It represents a stable and self-enforcing outcome, where no one has an incentive to deviate. However, a Nash equilibrium may not always be desirable or efficient, as it may involve suboptimal payoffs, conflicts, or dilemmas. For example, in the famous prisoner's dilemma game, the Nash equilibrium is for both players to confess and receive a harsh sentence, even though they would both be better off if they both remained silent.

3 Utility functions

A utility function is a mathematical representation of a player's preferences over the possible outcomes of a game. It assigns a numerical value to each outcome, reflecting the player's satisfaction, happiness, or welfare. A utility function can help you manage conflicting objectives in game theory by allowing you to compare and rank different outcomes, as well as to measure the trade-offs and risks involved. For example, in the auction game, a utility function can capture the player's willingness to pay, reservation price, and expected profit.

4 Dominant and dominated strategies

A dominant strategy is one that yields the best outcome for a player, regardless of what the other players do. A dominated strategy is one that yields a worse outcome for a player, regardless of what the other players do. Dominant and dominated strategies can help you simplify and solve games by eliminating strategies that are never optimal or always inferior. For example, in the chicken game, driving straight is a dominant strategy for both players, while swerving is a dominated strategy for both players.

5 Mixed and pure strategies

A pure strategy is one that involves choosing a specific action with certainty. A mixed strategy is one that involves choosing among several actions with some probability. Mixed and pure strategies can help you model and predict the behavior of players who face uncertainty, randomness, or incomplete information. For example, in the rock-paper-scissors game, a mixed strategy is to choose each action with equal probability, while a pure strategy is to choose one action with certainty.

6 Cooperative and non-cooperative games

A cooperative game is one in which the players can communicate, coordinate, or form binding agreements to achieve a common goal. A non-cooperative game is one in which the players cannot or do not cooperate, and act independently or competitively. Cooperative and non-cooperative games can help you understand and influence the incentives and outcomes of players who have conflicting or cooperative objectives. For example, in the public goods game, a cooperative game is one in which the players can agree on how much to contribute to the public good, while a non-cooperative game is one in which the players decide individually how much to contribute.

7 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?