Definition of mad situation

Let's imagine the following situation:
1) There is a game in which payoff functions are randomly generated for n roles in this game. Players can analyze these payoff functions and make independent moves that will determine the payoff in the current round.
2) There are 2 groups:
madmen who play each move completely randomly and
intelligent egoistic agents
3) Each group consists of n people such that all roles are occupied by people within each group.
4) Each group constantly plays more and more new rounds, so we estimate how big the payoffs of each player are on average.

The question is:
Are there payoff functions for which the madness strategy dominates the egoism strategy?
If so, how often do such payoff functions occur?

By "Independent Optimization Is Worse Than Randomness" and "the strategy of madness dominates the strategy of egoism" is meant that for any role in the game, the madman will on average earn more than the egoist.

Random Game Generator

Let's consider the simplest option:
1) 2 roles
2) Each player i chooses a number $x_{i}$ from the interval [-1., 1.]
3) Random functions are defined as:
$f_{i} (x_{1}, x_{2}) = c_{1}^{i} * x_{1} + c_{2}^{i} * x_{2}$ ,
where $c_{1}^{i}$ is uniformly chosen from [-1., 1.].

The probability of mad situations

We can easily calculate the average over x_i:
$M (f_{i} (x_{1}, x_{2})) = M (c_{1}^{i} * x_{1} + c_{2}^{i} * x_{2}) = c_{1}^{i} * M (x_{1}) + c_{2}^{i} * M (x_{2}) = 0$

Thus, for any such random function, lunatics will get 0 on average.

Let's consider ethereal agents, with independent optimization they will always get:
$x_{i} = s i g n (c_{i}^{i})$
$f_{1} (x_{1}, x_{2}) = c_{1}^{1} * s i g n (c_{1}^{1}) + c_{2}^{1} * s i g n (c_{2}^{2}) = | c_{1}^{1} | + c_{2}^{1} * s i g n (c_{2}^{2})$
$f_{2} (x_{1}, x_{2}) = c_{1}^{2} * s i g n (c_{1}^{1}) + c_{2}^{2} * s i g n (c_{2}^{2}) = c_{1}^{2} * s i g n (c_{1}^{1}) + | c_{2}^{2} |$

And we are interested in the probability with which these wins will both be less than 0.

For this we need and it is enough:
$s i g n (c_{2}^{1}) = - s i g n (c_{2}^{2})$ ,
$| c_{2}^{1} | > | c_{1}^{1} |$
$s i g n (c_{1}^{2}) = - s i g n (c_{1}^{1})$ ,
$| c_{1}^{2} | > | c_{2}^{2} |$

Since in our case the modulus of the number and its sign are independent random values, we have 4 independent conditions, each of which has a probability of $1 / 2$ .
And as an answer we get:
$p = (1 / 2)^{4} = 1 / 16$

Conclusion. Madness

When we talk about the prisoner's dilemma and things like that, we're talking about the fact that somewhere out there, there is a good state that is better for all players. This is not as mind-blowing as the fact that in some cases such states can be easily reached by simply replacing the egoists with madmen.
And the probability of such situations ...

Posts

Wikitag Contributions

Comments