Trembling hand perfect equilibrium

When Trembling -hand perfect equilibrium, there is a possibility of selection of Nash equilibria. It was developed by Reinhard Selten, the idea published in 1975 under the name "A Model of Slight Mistakes " in the " International Journal of Game Theory." The aim here is to determine how susceptible an equilibrium over a player's mistakes. After Rarely there is no error if the players act perfectly rational. However, in reality to be reckoned with wrong decisions of the opponent. To represent this in game theory, the Trembling -hand perfect equilibrium was one wound.

• 4.1 Example

Simple illustration of the approach

In simple terms, the idea of ​​trembling -hand- perfect equilibrium: Suppose player A believes that player B always plays the strategy and the best response of player A is on his strategy. Then strategy is always to play even then the optimal choice if Player B by mistake, talk with a low error probability of playing? If under these circumstances is still the best strategy of player A, so it is a Trembling -hand perfect strategy.

Trembling -hand perfect equilibrium in normal form games

By following normal form game, the approach of the Trembling -hand perfect equilibrium is very simple illustrated by the following payoff matrix:

The two Nash equilibria are in this example and. Now to be determined whether one of the two balances (or both) Trembling hand perfect equilibria. Suppose Player A wants to play his strategy and assumes that Player B will play his strategy, as both then would get a payoff of 3. However, Player A is not completely sure whether Player B but not with a low error probability plays his strategy. To find out if in spite of this error probability of player 2 the best choice of player 1, and thus is still Trembling -hand perfect, the following must be checked: The expected payoff of player 1 if he chooses must be at least as large as the expected payoff for the choice of.

Is the error probability of player B, which is assumed to be very small. That is, the counter probability associated with it. with

The expected payoff of player 1 for the choice of is:

In comparison, the expected payoff of the strategy:

It is easy to see that:

Even if player B plays with a small probability of error is the best possible choice for Player A. So the strategy is Trembling -hand perfect. A Trembling -hand perfect equilibrium is, however, a combination of two Trembling -hand perfect strategies. To examine therefore whether it is a Such strategy in combination, must be tested by player 2. Analogous to this strategy is as follows:

The expected payoff of player 2 for the choice of is:

In comparison, the expected payoff of the strategy:

And again, can clearly be seen that:

This means that the strategy is Trembling -hand perfect and it is the Nash equilibrium to a Trembling -hand perfect equilibrium.

Formal definition using a perturbed game

Simple definition of a perturbed game

A perturbiertes game is a copy of the underlying game, with the restriction that each player must play all pure strategies with positive probability. That is in the normal game, it is possible to Player A play his strategies. In the perturbed game must be.

Formal definition

The starting point was a game in strategic form:

Where the set of players is, the set of mixed strategies, which is based on the possible distribution of pure strategies and it is is the expected payoff of player. The central idea to map possible failures of the players, it is assumed that there is no pure strategy with a probability of zero can be played. In such a perturbed game is:

The amount of mixed strategies in the perturbed game is:

This means that players, each of his pure strategies of Quantity, must play at least with probability. It follows the perturbed game:

A Nash equilibrium in the perturbed game was and is a Nash equilibrium in the main game. If you let go in the perturbed game against zero probability of error and thus the balance of the perturbed game is to match as in the normal game, it is called a trembling -hand- perfect equilibrium. More formally, this means that if

Is it a trembling -hand- perfect equilibrium.

Example

As a starting game a normal form game is considered with the following payoff matrix:

In the normal game, there are two Nash equilibria, arising from the combination of strategies and. It comes with two players in their choice of each best answer depends on which strategy selects the respective opponent.

The perturbed game that is not the case. Because if there is only a tiny probability that player B chooses its strategy, then the best response of player A is to play. Since this probability in the perturbed game exists by definition, Player A would always play. However, Player A must in the perturbed game play its strategy with a positive probability. Therefore, its best response to play the mixed strategy - ie the smallest possible probability of.

Due to the symmetry of the game, the best strategy for player B is accordingly:

The equilibrium in the perturbed game is:

If you leave now and go against, then from the perturbed game almost back to the normal, because:

If you let go to zero the probability of error and thereby moved the balance of the perturbed game on a Nash equilibrium of the underlying game, then it is in this balance to a Trembling -hand perfect.

In this example, the perturbed game moves towards the equilibrium and thus Trembling -hand perfect.

Trembling -hand perfect equilibrium in sequential games

Also for sequential games, the concept of Trembling -hand perfect equilibrium is applicable. Similar to the normal form games, it is here for the selection of sub game Perfect equilibrium of benefits to determine whether such a balance is maintained even with a small probability of error.

Example

Im on the right side, there are four sub- game Perfect Equilibria:

Trembling -hand perfect equilibrium in extensive games with the agent normal form

Developed by game theorists Elon Kohlberg, and here slightly modified Dalekspiel, is an example of another application of the Trembling -hand perfect equilibrium. Developed in 1953 by Harold W. Kuhn agents normal form is applied at a reduction of an extensive game to the normal form, to compensate for the loss of information. As can be seen in Dalekspiel on the right, player 1 is divided into two agents to represent uncorrelated at each decision node decisions can mathematically.

In the normal form can be seen very easily that there are three Nash equilibria in pure strategies. To examine these equilibria now on their Trembling -hand perfectness, one must ensure that the error probabilities that Player 1 has at its two decision nodes are not correlated. This means that an error at the first decision node must not cause the probability of further errors is increasing or decreasing. To ensure this, player 1 is as described above, divided into two agents who meet independently of their decisions.

With the enrollees probabilities the normal form looks from above like this:

Now one can easily see that the strategies for player 1 and for small Trembling -hand perfect strategies. For he keeps a sure payoff of 2 no matter how player 2 chooses. In the only better payoff of 4 in the choice of he receives due to the probabilities in the expected value of the payout: