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# Decision rule

In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game theory.

In order to evaluate the usefulness of a decision rule, it is necessary to have a loss function detailing the outcome of each action under different states.

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• Decision Analysis 1: Maximax, Maximin, Minimax Regret
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#### Transcription

Welcome! In this brief video, we will be discussing decision making without probabilities. In this first part, we will consider Maximax or the optimistic approach, the maximin also known as conservative or pessimistic approach, and the minimax regret approach to decision-making. The table seen here is referred to as a payoff table or decision table. the alternate is on the left here in the rows are referred to as Decision Alternatives. They are the options available for the decision maker to choose from We will assume that the decision maker can only choose one of these alternatives - invest in bonds stocks, or mutual funds. In the columns we have the economic conditions. Since the decision maker does not have control over these, we refer to them as states of nature or outcomes. The values in the table are called payoffs. They could be profit, cost, distance, time, and so on. In this example we treat them as profits. The Maximax or Optimistic approach Using this optimistic approach, we choose the alternative with the best possible payoff. Looking at Bonds the best payoff is 45. The best is 70 for stocks, and the best is 53 for mutual funds. The overall best is 70. Therefore the decision is to invest in stocks. The maximin or conservative approach. Using this pessimistic approach we choose the alternative with the best of the worst payoffs. We first choose the the worst payoff in each alternative and then choose the best of the worst. Looking at Bonds, the worst payoff is 5, the worst is -13 for stocks and the worst is -5 for mutual funds. The best of these is 5. Therefore the pessimistic or conservative approach is to invest in bonds. The minimax regret approach. Using this approach which choose the alternative with the minimum of all maximum regrets across all alternatives. Regret, also known as opportunity loss is the difference between the best payoff in a particular state of nature and the actual payoff received. For example, if the economy is growing, the best payoff is 70. If we happened to have invested in bonds, then the regret will be 70 - 40 which is 30. If we invested in stocks then there is no regret. If we invested in mutual funds, then the regrets is 70 minus 53 which is 17. Again, if the economy stable, the best payoff is 45, so if we invested in bonds, there is no regret, the regret is 45 - 30 if we invested in stocks, if we invested in mutual funds there is also no regret. For declining economy the best payoff is 5. If we invested in bonds, there is no regret; if we invested in stocks the regret is five minus -13 which is 18; if we invested in mutual funds, the regret is 5 minus -5 which is 10. Here is the regret table. Since the decision is to be made based on minimax regret, we first determine the maximum regret for each alternative and then choose the minimum. For bonds, the maximum regret is 30. For stocks it is 18, and for mutual funds, it is 17. The minimum of these maximum regrets is 17. The decision is to invest in mutual funds. See you in part 2. Thanks for watching!

## Formal definition

Given an observable random variable X over the probability space ${\displaystyle \scriptstyle ({\mathcal {X}},\Sigma ,P_{\theta })}$, determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ : ${\displaystyle \scriptstyle {\mathcal {X}}}$→ A.

## Examples of decision rules

• An estimator is a decision rule used for estimating a parameter. In this case the set of actions is the parameter space, and a loss function details the cost of the discrepancy between the true value of the parameter and the estimated value. For example, in a linear model with a single scalar parameter ${\displaystyle \theta }$, the domain of ${\displaystyle \theta }$ may extend over ${\displaystyle {\mathcal {R}}}$ (all real numbers). An associated decision rule for estimating ${\displaystyle \theta }$ from some observed data might be, "choose the value of the ${\displaystyle \theta }$, say ${\displaystyle {\hat {\theta }}}$, that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose ${\displaystyle {\hat {\theta }}}$." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined, ${\displaystyle {\hat {\theta }}}$ could be chosen, for instance, using some optimization algorithm.
• Out of sample prediction in regression and classification models.