Games, Strategies, and Payoff Matrices

Game theory studies choices when outcomes depend on several decision makers. A game specifies players, available strategies, information, timing, and payoffs. Leaving one element vague can change the predicted behavior.

A strategy is a complete plan for the situations a player may face. An action is one move at one decision point. In a one-shot simultaneous game, the distinction is small; in a sequential or repeated game, a strategy must describe responses after different histories.

Reading a matrix

In a two-player matrix, rows belong to one player and columns to the other. A payoff pair $(a,b)$ normally gives the row player's payoff first and the column player's second.

To find a best response:

1. Hold the rival's strategy fixed.
2. Compare only your own payoff across your available strategies.
3. Mark every strategy tied for the highest payoff.
4. Repeat for every rival strategy and then for the other player.

Payoffs need not be money. They can summarize profit, time, risk, reputation, or policy goals using any numbers that preserve the player's ranking. They should not be added across players unless interpersonal comparison has a justified meaning.

Simultaneous and sequential games

A simultaneous game means players choose without observing the current move of the other player, not necessarily at the exact same second. Sequential games are often represented by decision trees and solved backward from later choices. Timing and observability can transform equilibrium.

Annotate one matrix completely

Suppose the platform chooses Promote or Standard and a restaurant chooses Join or Decline. If the restaurant joins, platform payoffs are 6 and 8, so Standard is its best response. If the restaurant declines, platform payoffs are 1 and 3, so Standard is again best.

Now compare the restaurant's second payoff across each row. It prefers Decline whether the platform promotes or stays standard. The cell Standard/Decline contains both best-response marks.

This procedure is deliberately mechanical: fix one rival action, compare one player's payoffs, mark all maxima, then switch players. Looking for the largest number anywhere or adding payoffs first will not identify best responses.

Write the game before solving it

Ask who moves, what each player knows, which actions are available, and what the payoffs represent. If the restaurant observes the platform's action before choosing, the matrix hides important timing and a decision tree is more appropriate. Strategy analysis cannot repair a badly specified game.