Minimax rule

The minimax rule (or maximin rule , occasionally also pessimism rule or Wald rule , according to Abraham Wald ) is a decision rule . It optimizes the result that can be reliably achieved, that is, the decision is based on the worst of all possible cases (the MINImum is MAXimized). This rule reflects a pessimistic attitude or the decision-making behavior of a risk-averse decision-maker. The opposite is the maximax rule .

Structure of the minimax principle

The structure of the minimax principle can be illustrated as follows.

Suppose you have three sets of numbers:

Set of numbers 1: ${\ displaystyle \ {5,29,44,200 \}}$
Set of numbers 2: ${\ displaystyle \ {11,17,34,39 \}}$
Of numbers 3 . ${\ displaystyle \ {4,310,1105,3451 \}}$

If one follows the minimax principle (“choose the set of numbers in which the smallest number is greater than the smallest number of any other set of numbers”), then the set of numbers 2 is chosen because the smallest number is “11” and “11” “11” is greater than the smallest number of the two remaining sets of numbers (the “5” and the “4”).

Making decisions at risk

The following example applies the minimax rule to decisions under risk .

A says to B:

You can roll the dice up to ten times.
If you don't roll a six, you won't get anything.
If you toss the six once, you get one euro.
If you toss the six twice, you won't get anything either.
If you toss the six three times, you get ten euros.

If the minimax rule were applied, B would roll the dice until he has rolled a six, because as long as he has not yet rolled a six, the result cannot worsen even in the worst case (if he does not roll a six), but it can improve ( if B rolls a six). Then B gets at least one euro instead of zero.

However, if he has thrown the first six, then B stops rolling, because it could happen that he throws the second six, but not the third. If this worst of all possible cases for B occurs, he would get nothing at all, while he would receive at least one euro.

As you can see, the minimax rule reduces the existing risk. It is in this sense that the rule is used in game theory . The minimax rule is used, for example, in two-person zero-sum games with perfect information , in which one wins what the other loses and vice versa.

Decisions in conflict situations

In conflict situations, the minimax rule can help to settle on a common basic level, e.g. B. in negotiations to agree. The prerequisite for this is a game with perfect information, which means that each side knows the other side's points, "the cards are on the table". If the Nash equilibrium is used here, the optimal result is achieved with which both sides can do something without the parties having to regret their decision. In any case, the risk is as low as possible and thus an excellent basis for further cooperation.

Support in difficult development processes

If the goal is achieved, the minimax rule will help to implement plans and goals. The reason for this lies in the path of least resistance. Here the minimax rule no longer has anything to do with pessimism or zero-sum play. Resistance to a thing, for example, is then rated as a so-called payoff. You don't need a second player here either.

Collective decisions

The minimax rule can also be applied to collective decisions. This has z. B. John Rawls done in his theory of justice . Then the minimax rule is:

"Collectively chosen is that alternative in which the worst-off individual is still better off than any of those individuals who are worst-off when the other alternatives occur."

Assume a group, consisting of the individuals , and are faced with the decision between the alternatives , and , where the numbers in the table denote the quantities of some good - e.g. B. Vacation days. Each individual would rather have more than less of the good: ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle C}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$

	A.	B.	C.	${\ displaystyle \ min _ {i} U (E_ {k, i})}$
x	3	3	3	3
y	8th	2	10	2
z	4th	5	6th	4th

When applying the minimax rule to the values in the table above, the alternative is chosen collectively, because in this case the worst-off individual with four vacation days is still better off than the worst-off in the case of the other two alternatives (at there are three and two vacation days). ${\ displaystyle z}$ ${\ displaystyle A}$ ${\ displaystyle x}$ ${\ displaystyle y}$

Ordinal measurement level

The minimax rule works with evaluations of the alternatives in the form of rankings, so only requires an ordinal measurement level of the individual values. The minimax rule, however, requires an interpersonal comparison of individual welfare levels (eg: "A is better off than B"). In our case it is assumed that vacation days have the same value for all individuals.

Dependence on the way the decisions are bundled

One problem with the minimax rule is its dependence on the way decisions are bundled. The Minimax rule shares this problem with other decision rules that only work with preferences and ratings in the form of rankings, such as: B. the majority principle.

Suppose the three individuals A, B and C have three separate decisions to make between two alternatives, s or t, v or w and x or y.

The alternatives correspond to certain fictitious quantities of any good (e.g. vacation days) that individuals have to add or give up when they choose the respective alternative collectively. It is assumed that each individual prefers owning a larger amount of this good to a smaller amount.

Three individuals make three joint decisions between two alternatives each

	A.	B.	C.
s	1	2	2
t	0	5	5
v	2	1	2
w	5	0	5
x	2	2	1
y	5	5	0

As can be seen from the table, in the case of separate decisions according to the minimax rule, the alternatives s, v and x would be chosen collectively.

However, the following table shows that the bundle of alternatives t + w + y is preferred to the bundle of alternatives s + v + x by all parties involved.

Three individuals make a decision between two bundles of alternatives

	A.	B.	C.
s + v + x	5	5	5
t + w + y	10	10	10

Such suboptimal results usually occur when the minimax rule is applied to a series of independent decisions when the individuals are in the decisive minimax position for the individual decisions that are less important to them and are not taken into account in the decisions that are important to them.

Even according to the minimax rule, a decision would be made between the two bundles of alternatives t + w + y and not as with the individual decisions s, v and x.

Minimax rule and the prisoner's dilemma

The minimax rule can be applied to the so-called prisoner's dilemma . This illustrates the connections to other essential terms in game theory.

	engage	Strolling
engage	3.3	1.4
Strolling	4.1	2.2

How will the players behave if they play according to the minimax strategy? You will choose the strategy of “strolling”. This realizes the Nash equilibrium, i.e. none of the players has an incentive to deviate from it. However, the pair of strategies (engage, engage) would mean a better outcome for both players. The minimax rule is pessimistic and in no way optimizes the payouts. The payout (3.3) could be realized if both players tried to maximize their maximum profit per strategy (opposite approach of the minimax rule).

literature

Gérard Gäfgen : Theory of Economic Decision. Investigations into the logic and economic importance of rational action. Mohr, Tübingen 1963 (at the same time: Cologne, university, habilitation paper, 1963).

Individual evidence

^ Henry Schäfer: Corporate Investments. Basics in theory and management. 2nd, revised edition. Physica Verlag, Heidelberg 2005, ISBN 3-7908-1580-2 , p. 231.
^ Andreas Diekmann: Game theory. Introduction, example, experiments . 3. Edition. rowohlts encyclopedia, Reinbek near Hamburg 2013, ISBN 978-3-499-55701-9 , p. 262 .

[1] Henry Schäfer: Corporate Investments. Basics in theory and management. 2nd, revised edition. Physica Verlag, Heidelberg 2005, ISBN 3-7908-1580-2 , p. 231.

[2] Andreas Diekmann: Game theory. Introduction, example, experiments . 3. Edition. rowohlts encyclopedia, Reinbek near Hamburg 2013, ISBN 978-3-499-55701-9 , p. 262 .