Game Theory Study notes (10) hybrid strategy

Definition of a hybrid policy

Assuming that pi represents the probability of adopting each pure strategy, we use PI (SI) to represent the probability that the participants adopt the strategy Si under the mixed strategy Pi, i.e. pi (SI) is the probability that pi gives pure strategy si.
For example, in the game of scissors and stone cloth, for participants i,pi= (1/3,1/3,1/3), the probability of choosing a rock is Pi (R) =1/3.
A hybrid strategy can be a pure policy.
The expected benefit of the hybrid strategy pi is the weighted average of the expected gain for each pure strategy in the hybrid strategy.

The corresponding games are: Tennis games, baseball games, dating and tax issues. Here is the main introduction of tennis game.

Tennis questions

Strategies and benefits for attackers and defensive parties:

Conclusion: There are two pure strategy Nash equilibria: (l,l) and (R,r). In addition there is a hybrid strategy for Nash equalization: for participants 1,u1 (l) = 2 * q + 0 * (1-q) = 2 * QU1 (R) = 0 * q + 1 * (1-q) = 1-q at Nash Equalization, U1 (L) = U1 (R), q = 1/3 for participating 2,U2 (L) = 1 * p + 0 * (1-p) = PU2 (r) = 0 * p + 2 * (1-p) = 2-2 * p at Nash Equalization, U2 (L) = U2 (r), p = 2/3 when q is OK, regardless of the probability of Participant 1 choice, its benefits are Fixed. Similarly, when P is determined, the benefit is fixed regardless of the probability that the participant 2 chooses.

If a hybrid strategy is the best strategy, then every pure policy in his strategy is also the best strategy.
After a mix-and-measure approach to Nash equilibrium, changing your own strategy gains will be constant, as the benefits depend on how often your opponent chooses each strategy.

Game Theory Study notes (10) hybrid strategy