Probability and Strategies

Introduction

The concept of strategy, understood as the ability to achieve a given objective by providing the means to do so, is fundamental in all aspects of daily life; all the more so when not all the elements in our possession can be determined with certainty. Numerous experiments have shown that mice also adopt a refined behavioral model based on the calculation of probabilities when it comes to waiting for a reward. The literature on the subject of strategies related to random games is very extensive, for a particular study of Stochastic and Bayesian Games you can refer to the text by Leyton-Brown, & Shoham (2008).

Strategy plays an important role during the games. In a game there are three key concepts:

• 1.

  the scenario, i.e., the environment in which the game takes place;

• 2.

  the randomness, which intervenes to a greater or lesser extent in most games;

• 3.

  the bet, which may be absent or present as in gambling.

Two cases can be distinguished here: Pure strategy completely defines the way a player plays a game, so what choice he will make in each situation he can find; Mixed strategy is a distribution of probability over all the available ones. In practice, a player can choose any pure strategy with 0 ≤ p ≤ 1.

A particular strategic game in its simplicity is skunk, proposed in a didactic key by Dan Brutlag (1994) but originally launched in 1953 by William Herbert Schaper as a board game in a box. The term indicates the animal but, in slang, means to defeat heavily, annihilate the opponent. Before starting the game each competitor has a sheet with five columns on which to write the results (Table 1).

The object of the game is to accumulate points by rolling two dice. The points are accumulated by making several throws, but by choosing to stop before the face with the value one comes out on one of the dice (a skunk in the original version). In this situation, the points of the turn are deleted. If a double one is rolled, then all the points accumulated in the previous columns are deleted.

The game itself is not complex, but it can be a starting point to answer some questions that imply a reflection on some not secondary aspects. What are the odds of throwing two dice, subject to certain constraints? Was the winner just lucky? Which strategy should be used in the game?

If we try to analyze the space of events, it is easy to see that the probability of double one is 1/36, that of simple one is 10/36 and that a favorable throw has a probability of 25/36. If we multiply each positive result by the probability of its achievement we see that the average score is 8 (Table 2).