Definitions
1. Pure Nash Equilibrium
It is when there is a state in which no player can make better profit by changing their action guaranteed no one else will change unreasonably their choice (other players are logical players)
2. Mixed Nash Equilibrium
It is when you don’t know the opponents movement, or it is a probability distribution, then you have a mixed Nash equilibrium
3. Dominated Actions
It is if doing one action instead of another one is stupid, it is a dominated action. For example assume player A and B has maximizer matrices (they want to maximize the outcome) as follows:
Here it doesn’t make any sense for player 1 to play the second action as it leads to 0 in each case (left) which is dominated by action 1 and action 3. Notice, action 3 dominates all outcomes because in every case (no matter it’s 4 or 5), we get a higher return compared to both 1 and 0.
4. Security Level and Security Policy
Security level is the worst possible outcome for ANY action taken by the opponent if you select a security policy. Naturally, a security policy is the action that produces the best security level value in case of any action taken by your opponent.
For the above example(in section 3), the security level for player 1 is 4 where the security policy is to take action 3. The minimum she can get is 4(right/bottom) when she plays action 3.
Note: The other player may not be adversarial and some actions might not be logical to be made by the other player. We just neglect this effect here.
Mixed Strategies
We only have a probability distribution of the actions we and our opponents will make. Therefore security level and security policy are actually minimizers / maximizers of total outcome formed by all possible action pairs (both your actions and your opponent’s). The probability distribution is optimized.
Equilibrium
We still have an equilibrium here. However, the equilibrium is not a single state, but a mixture of different states. For example, if the game is heads and tails thrown at the same time and one would win depending on the coins if they are same/different; one can do nothing but play randomly. So, there is no specific state in this strategy, but only a equilibrium probability distribution[1].
In detail, we have the players optimizing their actions just to make sure the other player never gets a clear logical move. For example[4]:
Notice the first rule, EU_L = EU_R. The meaning of this rule is actually obvious. If you make all of the options equally choosable for your opponent, your opponent can’t simply choose a single strategy, but need to select a random one again.
For example, if we say EU_L = 0, than we will see EU_R is 0.5 for the red player, and red player will play always to the right producing sometimes 2, but sometimes 0. In total, the red player will win in such a game.
References
[1] https://www.youtube.com/watch?v=fvEQujUcPv4
[2] https://ocw.mit.edu/courses/economics/14-11-insights-from-game-theory-into-social-behavior-fall-2013/study-materials/MIT14_11F13_Pure_strategy.pdf
[3] https://saylordotorg.github.io/text_introduction-to-economic-analysis/s17-03-mixed-strategies.html
[4] https://www.youtube.com/watch?v=aa8USttcDoE
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