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Session Recording

{% youtube %}https://youtu.be/8sqATVZD1S4{% endyoutube %}

We went over a bunch of different examples of mixed strategies :)

3.1

Instead of going through the book which for this section at least seems super complex, I'm using https://www.youtube.com/watch?v=FU6ax5K9HIA

  • Strategy: strategy s_i is for player i, any probability distribution over actions A_i

  • Pure strategy: Only one action is chosen with positive probability

  • Mixed strategy: Players introduce an element of randomness to their strategy, they can choose multiple actions and give them postiive probability

  • Support of mixed strategy: set of actions which are played with postive probability

  • Set of all strategies for i be S_i

  • Set of all strategy profiles be S = S_1 x ... x S_n

  • Because we changed the definition of strategies to include an infinite set of probability distributions, we have to change our utility functions to instead represent expected utility as opposed to the percice utility.

  • New utility definition:

screen shot 2016-12-13 at 5 16 21 pm

  • ENGLISH EXPLANATION: This new utility represents the sum of all of action profiles, considering each action profiles' utility multiplied by the probability that that action profile will be reached given a particular mixed strategy profile. The 2nd equation defines the probability of a particular action profile given a strategy profile as the product of each player playing that action profile.
  • We can now reevaluate our definition of best response and nash equilibrium:

screen shot 2016-12-13 at 5 24 33 pm

  • Best response in English: s*_i is a member of the set of best responses if and only if: for all s_i contained in S_i, the utility of playing s*_i while everyone else plays s_-i is at least as good as the utility of playing s_i while everyone else plays s_-i
  • Nash Equlibrium in English: Strategy s is a Nash equilibrium if and only if for all players (agents), everyone is playing a best response.
  • Awesome Thereum: Every mixed strategy finite game has a Nash equilibrium using

Exercises

Battle of the Sexes: Simple solution

This video shows a simple method by which you can solve for nash equilibrium in small mixed strategy games in which you can guess the support. I'm happy to go over this.

35.1

Same as pure strategy nash because everyone would change their probabilities to lower and lower until they all reach 1

35.2

(Incorrect solutions to both exercises: https://cloud.githubusercontent.com/assets/706123/21193487/778b496e-c1fa-11e6-9b55-1e4ba0f7deaa.jpg )

36.1

Totally incomplete solution, basically just a sad attempt at a mathematical formulation of the fact that the utility of each player's actions has to be equal as long as it is part of the support. https://cloud.githubusercontent.com/assets/706123/21194071/8f755ebe-c1fc-11e6-809b-9cf3afe59d49.jpg