Payoff Matrix

The Payoff Matrix is an expression of the First Law of Decision Science.

Each row represents one action that the decision maker might or might not freely choose to perform;

Each column represents a possible state of nature. At the time the decision must be made the decision maker assumes that one of the columns represents the actual decision situation, but her or she does not know which column is the correct one.

The cells of the matrix represent payoffs that the decision maker would receive if he or she chose the action represented by a particular row and the actual state of nature were the one represented by a particular column.

Two Kinds of "Easy" Decisions

Certainty: When we make a decision under the assumption of certainty, we only consider one state of nature to be possible; the payoff matrix has only one column of payoffs, and the best decision is the one with the best payoff. Linear programming is an example of a technique for making decisions under the assumption of certainty.

Stochastic Dominance: If action A has a better payoff than action B under each individual state of nature, then we say that action B is stochastically dominated by action A. If the payoff matrix truly represents every thing the decision maker hopes (or fears) to receive from the decision in question, then no rational decision maker will ever choose to perform action B.

What is a Decision Rule?
In the general case, we must decide among two or more alternative actions that are not stochastically dominated, and more than one state of nature is considered possible. The payoff matrix assigns each alternative action a row of possible payoffs. A decision rule converts this row of payoffs into a single number that somehow represents the whole row. Once this is done, it is easy to pick the alternative action whose representative number is better than the other alternative actions' representative numbers.

Several decision rules exist; each has a different way to convert a row of possible payoffs into a single representative number-- for example:

  • the simple average of the possible payoffs for an alternative actions,
  • the worst possible payoff for an alternative action,
  • the best possible payoff for an alternative action,
  • the statistical expectation of the possible payoffs for an alternative action,
  • the maximum possible regret for an alternative action,
  • the statistical expectation of the possible regrets for an alternative action.