Wisc ID for in-class quiz: (if your wisc email is "test@wisc.edu", please enter "test")
Token: (will be given during the lectures) 20 id,answer_id;token,answer_check
# Warning: this is a draft and will be updated one day before the lecture.
📗 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.
In-class 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: .
In-class 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.
In-class 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.
📗 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: 0).
📗 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
\(-1\)
\(1\)
\(0\)
In-class 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): .
In-class 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?
📗 If you have questions, please use (i) Zoom chat, (ii) Piazza: Link, (iii) Office hours and discussion sessions. Please do NOT use Canvas mail and use email only to the course instructor (not TAs) for grading issues.
Additional In-class Discussion
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Additional In-class Quiz
📗 Sometimes a question not in the notes will be asked during the lecture, you can submit your answer here:
A.
B.
C.
D.
E.
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📗 To get full points on the in-class quizzes for a lecture:
➩ Submit relevant answers to the questions discussed during the lecture: incorrect answers are okay.
➩ Some questions require [notes] to earn the point.
➩ Some questions require special ID (given during the lecture) to earn the point.
➩ Do not submit answers to questions that are not discussed during the lectures. Each such submission will result in a deduction of one point.
➩ Submissions after the lecture, before the midterm (first 14 lectures) and the final exam (last 14 lectures), are accepted. After the exams, no in-class quiz submissions will be accepted.
➩ The grade on Canvas Assignment Q20 is computed as number of points divided by the number of questions asked (out of 1) and updated on Canvas every weekend.
📗 If there are any issues with submission on the website, please use this Google form: Link.
📗 Bonus point opportunities during a few lectures (added to in-class quiz above 20 points).
📗 Notes and code adapted from the course taught by Professors Jerry Zhu, Blerina Gkotse, Yudong Chen, Yingyu Liang, Charles Dyer. Some content are generated using Copilot .