# Choice under Uncertainty

Uncertainty means that the probability of a certain event is not known. To choose an option, different methods are available:

- Maximin rule
- Maximax rule
- Range rule
- Hurwicz rule
- Minimax-Regret rule
- Laplace Rule

### Maximin Rule

This rule is the most pessimistic one. It assumes that the worst case happens. And it tries to maximize the pay-off in this situation. In short, we choose the option that grants us the highest (maximal) minimal pay-off.

Suppose we have the following table with three alternatives A and their pay-offs in the situations S 1 till 4

S_{1} | S_{2} | S_{3} | S_{4} | Min | |

A_{1} | 20 | 15 | 20 | 3 | 3 |

A_{2} | 5 | 6 | 7 | 4 | 4 |

A_{3} | 22 | 3 | 3 | -2 | -2 |

The minimal pay-off for A_{1} is 3, for A_{2} it is 4 and for A_{3} it is -2. Now we choose the alternative that yields us the highest of these pay-offs. This is A_{2}.

### Maximax Rule

The Maximax rule is the most optimistic one. It assumes that the best case happens. And it tries to maximize the pay-off in this situation. In short, we choose the option that grants us the highest (maximal) maximal pay-off.

S_{1} | S_{2} | S_{3} | S_{4} | Max | |

A_{1} | 20 | 15 | 20 | 3 | 20 |

A_{2} | 5 | 6 | 7 | 4 | 7 |

A_{3} | 22 | 3 | 3 | -2 | 22 |

The alternative with the highest pay-off is A_{3}.

### Range Rule

In the range rule, we assign ranks for each alternative in each Situation. Then, we sum up the ranks and choose the alternative with the best rank.

Suppose we have the following payoff table with two alternatives and 100 situations.

S_{1} | S_{2} | S_{3} | … | S_{100} | |

A_{1} | 0 | 3 | 3 | 3 | 3 |

A_{2} | 1 | 1 | 1 | 1 | 1 |

The Maximin rule would tell us to choose A_{2}. But this sounds unintuitive because there are 99 situations that would bring a better result.

Now, we assign a rank. The lower the rank, the better. In this example, our lowest rank is 0. (But it is also possible to use 1 as the lowest rank.)

| S_{1} | S_{2} | S_{3} | … | S_{100} | Sum |

A_{1} | 1 | 0 | 0 | 0 | 0 | 1 |

A_{2} | 0 | 1 | 1 | 1 | 1 | 99 |

Then, we choose the alternative with the lowest total rank. In our case this is A_{1}.

### Hurwicz Rule

The Hurwicz rule introduces an optimism parameter a. The rank for each alternative is calculated by:

H = (1-a)*Min + a*Max.

Min: Minimal pay-off of an alternative

Max: Maximal pay-off of an alternative

Suppose we have the following table and choose an a = 0.4.

S_{1} | S_{2} | S_{3} | S_{4} | Min | Max | H | |

A_{1} | 20 | 15 | 20 | 3 | 3 | 20 | 3*(1-0.4)+20*0.4 = 9.8 |

A_{2} | 5 | 6 | 7 | 4 | 4 | 7 | 5.2 |

A_{3} | 22 | 3 | 3 | -2 | -2 | 22 | 7.6 |

Alternative A_{1} has the highest H. That’s why we would choose it.

### Minimax Regret Rule

With this decision rule, we try to minimize the regret of a choice. Regret is the difference between the maximal possible pay-off in a situation and the realized pay-off.

R(a_{k},s_{j}) = max U(s_{j}) – U(ak,s_{j})

- Find for every situation s
_{j}the option with the maximal pay-off. - Calculate for every option in situation s
_{j}the maximal loss. - Find for each option the maximal loss from in all situations.
- Choose the option with the smallest loss R.

S_{1} | S_{2} | S_{3} | S_{4} | |

A_{1} | 20 | 15 | 20 | 3 |

A_{2} | 5 | 6 | 7 | 4 |

A_{3} | 22 | 3 | 3 | -2 |

Max | 22 | 15 | 20 | 4 |

S_{1} | S_{2} | S_{3} | S_{4} | Rmax() | |

A_{1} | 22-20 = 2 | 15-15 =0 | 0 | 1 | 2 |

A_{2} | 22-5 = 17 | 15-9 =6 | 13 | 0 | 17 |

A_{3} | 22-22 =0 | 15-12 =3 | 17 | 6 | 17 |

A_{1} is the option with the smallest maximal regret.

### LaPlace Rule

Now, we assume that every situation occurs with the same likelihood. Then we choose the option with the highest expected pay-off.

S_{1} | S_{2} | S_{3} | S_{4} | Expected pay-off | |

A_{1} | 20 | 15 | 20 | 3 | (20+15+20+3)/4=14.5 |

A_{2} | 5 | 6 | 7 | 4 | 5.5 |

A_{3} | 22 | 3 | 3 | -2 | 6.5 |

Option A_{1} has the highest expected value.