Minimax Principle: Decision Criterion for Minimizing Regret

The minimax principle is a decision criterion aimed at minimizing the worst-case scenario, thus reducing possible regret by ensuring the most unfavorable outcome is as favorable as possible. It finds extensive applications in decision theory, game theory, and economics.

The minimax principle (short for minimization of maximum loss) is a fundamental decision criterion that emerges significantly in decision theory and game theory. By focusing on minimizing the worst-case scenario, the minimax principle seeks to reduce the “regret” associated with decision-making under uncertainty. It ensures that even the worst outcome of a decision is optimized to be as favorable as possible.

Key Concept

At the core of the minimax principle is the concept of regret, or the difference between what is achieved and the best possible outcome that could have been achieved had the circumstances been different.

Mathematically, if a decision-maker is evaluating a set of decisions \(D\) with possible outcomes \(O\), the minimax principle involves:

$$ \min_{d \in D} \max_{o \in O} L(d, o), $$

where \(L(d, o)\) represents the loss associated with decision \(d\) and outcome \(o\). The aim is to choose a decision \(d\) such that the maximum loss (worst-case scenario) is minimized.

Applications in Various Fields

Game Theory

In game theory, the minimax principle is crucial for determining optimal strategies in zero-sum games. Each player formulates a strategy to minimize the maximum loss inflicted by an opponent. The concept is foundational in formulating Nash Equilibria and finding optimal mixed strategies.

Economics

In economics, the minimax criterion is used in scenarios where market participants face uncertain outcomes. It often applies to investment strategies, insurance policies, and market exit decisions where the least favorable outcome is optimized.

Decision Theory

In decision theory, managers and policymakers use the minimax approach to make decisions that guard against the worst possible misfortunes. This includes scenarios such as crisis management, resource allocation under uncertainty, and competitive strategy in uncertain markets.

Example of Minimax Principle

Consider an investor choosing between two investment options (A and B) with uncertain returns described by the following payoff matrices:

State of Economy	Investment A	Investment B
Recession	-50	-100
Stable	100	50
Boom	200	400

Applying the minimax principle:

• Maximum loss for Investment A = -50
• Maximum loss for Investment B = -100

According to the minimax principle, the investor would choose Investment A as it minimizes the worst loss scenario.

Historical Context

Origins and Development

The minimax principle has its roots in the work of mathematician John von Neumann, who formulated the principle in the context of two-player zero-sum games. It was further developed and formalized in the early 20th century, becoming an essential tool in operations research and economic theory.

Applicability

The principle applies not only in competitive environments but also in collaborative and uncertain scenarios where decision-makers aim to safeguard against unfavorable outcomes. Its adoption has expanded from theoretical economics and game theory to practical domains like engineering, healthcare, and artificial intelligence for robust decision-making models.

Related Terms

• Maximin Principle: A closely related concept is the maximin principle, where the decision-maker seeks to maximize the minimum gain. This is the converse to the minimax principle and applies in contexts where ensuring a baseline positive outcome is crucial.
• Nash Equilibrium: The Nash equilibrium describes the situation where no player can benefit by unilaterally changing their strategy. Often, the strategies derived involve principles from minimax optimization, particularly in zero-sum games.

FAQs

What is the Minimax Principle used for?

The minimax principle is used for making decisions under uncertainty to minimize the worst-case loss or regret, frequently employed in game theory, economics, and decision theory.

How does the Minimax Principle differ from the Maximin Principle?

While the minimax principle minimizes the worst loss, the maximin principle maximizes the minimum gain, focusing on ensuring a baseline positive outcome.

Is Minimax used in Artificial Intelligence?

Yes, the minimax algorithm is widely used in artificial intelligence, especially in game-playing bots like chess and tic-tac-toe, to minimize the possible loss against an opponent’s best strategies.

References

• von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
• Savage, L. J. (1954). The Foundations of Statistics. Dover Publications.

Summary

The minimax principle is a vital decision criterion for minimizing the worst-case scenario or regret associated with decisions under uncertainty. Its applications extend across game theory, economics, decision theory, and artificial intelligence, offering a robust framework for optimal decision-making when faced with the potential of unfavorable outcomes. By focusing on the minimization of maximum loss, the minimax principle guides individuals and organizations towards safer, more rational choices in complex environments.