Max Min Strategy
How to choose an alternative?
Idea: Extreme Pessimism
- Base your choice on the worst situation that can happen
- and maximize the consequences in the worst case
- $\max_{a \in A} \min_{e \in E} c(a, e)$
In other words:
- We don’t know what will be the state of nature (or the strategy played by the other player)
- But since we’re pessimistic, we choose the worst outcome
- and we want to select the alternatives that are best in the worst case
Example 1: Decision Under Uncertainty
Consider the following example:
| $c$ | $e_1$ | $e_2$ | $e_3$ | min | $a_1$ | 40 | 70 | -20 | -20 | worst case for $a_1$ | $a_2$ | -10 | 40 | 100 | -10 | worst case for $a_2$ | $a_3$ | 20 | 40 | -5 | 5 | worst case for $a_3$ | max: | 5 | $\to a_3$ |
In the “min” column shows the worst cases for all alternatives
- now we select the maximal value from them
- and base our decision on it
Advantages and Disadvantages
Downsides
- bad use of information
- we have a lot of information in the table, but not using only one value from each row
- and the choice is based only on one value
- no compensation between different consequences
- see an example below
No Compensation
| $c$ | $e_1$ | $e_2$ | … | $e_{1000}$ | min | $a$ | -100 | 1000 | 1000 | … | 1000 | -100 || $b$ | -99 | -99 | -99 | … | -99 | -99 || | | | | | | ‘’‘-99’’’ | In this case we choose $b$
- although it’s clear that $a$ is better at everything
- it’s only a little little bit worse at $e_1$
Advantages:
- no need for rich scale:
- enough to have only an ordinal scale
- only need to rank the alternatives and take the worst/the best
Max Min and Min Max
; maxmin strategy
- strategy that maximizes player’s worst-case payoff
- when others want to cause the greatest harm
- i.e. it maximizes the minimal outcome
- maximin value - guaranteed minimal payoff
; minmax strategy
- of player $i$ against player $-i$
- one that minimizes the maximal value of $-i$
; minimax theorem
- in any finite 2-players zero-sum game
- each player receives a payoff that is equal to both
- his minimax and his maximin value
- 0 is a saddle point
- <img src=”
” /> - '’saddle point’’ - where minimax = maximin