📗 Unlike sequential games, for simultaneous move games, one player (agent) does not know the action taken by the other player.
📗 Given the actions of the other players, the optimal action is called the best response.
📗 An action is dominated if it is worse than another action given all actions of the other players.
➩ For finite games (finite number players and finite number of actions), an action is dominated if and only if it is never a best response.
➩ An action is strictly dominated if it is strictly worse than another action given all actions of the other players. A dominated action is weakly dominated if it is not strictly dominated.
📗 Rationalizability (IESDS, Iterative Elimination of Strictly Dominated Strategies): iteratively remove the actions are that dominated (or never best responses for finite games): Wikipedia.
TopHat Discussion
📗 [1 points] Write down an integer between and that is the closest to two thirds \(\dfrac{2}{3}\) of the average of everyone's (including yours) integers.
📗 Answer: .
TopHat Quiz
(Past Exam Question) ID:
📗 [4 points] Perform iterated elimination of strictly dominated strategies (i.e. find rationalizable actions). Player A's strategies are the rows. The two numbers are (A, B)'s payoffs, respectively. Recall each player wants to maximize their own payoff. Enter the payoff pair that survives the process. If there are more than one rationalizable action, enter the pair that leads to the largest payoff for player A.
📗 Another solution concept of a simultaneous move game is called a Nash equilibrium: if the actions are mutual best responses, the actions form a Nash equilibrium: Wikipedia.
TopHat Quiz
(Past Exam Question) ID:
📗 [4 points] What is the row player's value in a Nash equilibrium of the following zero-sum normal form game? A (row) is the max player, B (col) is the min player. If there are multiple Nash equilibria, use the one with the largest value (to the max player).
📗 A symmetric simultaneous move game is a prisoner's dilemma game if the Nash equilibrium (using strictly dominant actions) is strictly worse for all players than another outcome: Link, Wikipedia.
➩ For two players, the game can be represented by a game matrix: \(\begin{bmatrix} - & C & D \\ C & \left(x, x\right) & \left(0, y\right) \\ D & \left(y, 0\right) & \left(1, 1\right) \end{bmatrix}\), where C stands for Cooperate (or Deny) and D stands for Defect (or Confess), and \(y > x > 1\). Here, \(\left(D, D\right)\) is the only Nash equilibrium (using strictly dominant actions) but \(\left(C, C\right)\) is strictly preferred by both players.
Example
📗 Split or Steal games: (YouTube playlist: Link, Solution: Link).
📗 A mixed strategy is when a player randomizes between multiple actions: Wikipedia.
📗 A pure strategy is when a player uses only one action with probability 1.
📗 A mixed strategy Nash equilibrium is a Nash equilibrium for the game in which mixed strategies are allowed (called mixed extension of the original game).
➩ If the mixed strategies are mutual best responses, then they form a mixed strategy Nash equilibrium (see Math Note for the mathematical definition).
Math Note
📗 For a two-player general-sum game with rewards \(r\) (both players are maximizing), an action profile \(\left(a_{1}, a_{2}\right)\) is a pure strategy Nash equilibrium if \(r_{1}\left(a_{1}, a_{2}\right) \geq r_{1}\left(a', a_{2}\right)\) for every \(a'\) valid action of player \(1\); and \(r_{2}\left(a_{1}, a_{2}\right) \geq r_{2}\left(a_{1}, a'\right)\) for every \(a'\) valid action of player \(2\).
📗 An mixed strategy profile \(\left(\pi_{1}, \pi_{2}\right)\), where \(\pi_{i}\left(a\right)\) is the probability that player \(i\) chooses action \(a\), is a mixed strategy Nash equilibrium if \(\mathbb{E}\left[r_{1}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{1}\left(\pi', \pi_{2}\right)\right]\) for every \(\pi'\) mixed strategy of player \(1\); and if \(\mathbb{E}\left[r_{2}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{2}\left(\pi_{1}, \pi'\right)\right]\) for every \(\pi'\) mixed strategy of player \(2\).
➩ Here, the expectation is over the randomness of the mixed strategies, or \(\mathbb{E}\left[r_{i}\left(\pi_{1}, \pi_{2}\right)\right] = \displaystyle\sum_{a_{1}, a_{2}} r_{i}\left(a_{1}, a_{2}\right) \pi_{1}\left(a_{1}\right) \pi_{2}\left(a_{2}\right)\), where the sum is over all valid \(\left(a_{1}, a_{2}\right)\) actions of players \(1\) and \(2\).
➩ This condition is usually difficult to check: another equivalent condition is: \(\left(\pi_{1}, \pi_{2}\right)\) is a mixed strategy Nash equilibrium if \(\mathbb{E}\left[r_{1}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{1}\left(a', \pi_{2}\right)\right]\) for every \(a'\) valid action of player \(1\); and if \(\mathbb{E}\left[r_{2}\left(\pi_{1}, \pi_{2}\right)\right] \geq \mathbb{E}\left[r_{2}\left(\pi_{1}, a'\right)\right]\) for every \(a'\) valid action of player \(2\).
Example
📗 For rock paper scissor game, there is no pure Nash equilibrium, but there is one mixed Nash equilibrium where every player uses each action with probability \(\dfrac{1}{3}\): Link.
📗 The game matrix is:
\(a_{1} \setminus a_{2}\)
Rock
Paper
Scissors
Rock
\(0\)
\(-1\)
\(1\)
Paper
\(1\)
\(0\)
\(-1\)
Scissors
\(-1\)
\(1\)
\(0\)
TopHat Quiz
(Past Exam Question) ID:
📗 [4 points] Given the following game payoff table, suppose the row player uses a pure strategy, and column player uses a mixed strategy playing L with probability \(q\). What is the smallest and largest value of \(q\) in a mixed strategy Nash equilibrium?
Row \ Col
L
R
U
D
Note: the following is a diagram of the best responses (make sure you understand what they are and how to draw them). The red curve is the best response for the column player and the blue curve is the best response for the row player.
📗 Answer (comma separated vector): .
TopHat Quiz
(Past Exam Question) ID:
📗 [4 points] Given the following BoS (Battle of Sexes) game, what is the row (Romeo) player's (expected) value (i.e. payoff) in the mixed strategy Nash equilibrium?
📗 Every finite game has a (possibly mixed) Nash equilibrium: Wikipedia.
testtt,ie,zs,cm,bosq
📗 Notes and code adapted from the course taught by Professors Jerry Zhu, Yudong Chen, Yingyu Liang, and Charles Dyer.
📗 Content from note blocks marked "optional" and content from Wikipedia and other demo links are helpful for understanding the materials, but will not be explicitly tested on the exams.
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