📗 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: .
[Note] Use the space to explain the steps or just take notes:
[L20Q]
Other students' answers:
In-class Discussion
📗 [1 points] Two identical antiques are lost. The airline only knows that its value is at most dollars, so the airline asks their owners (travelers) to report its value (non-negative integers larger than or equal to . The airline tells the travelers that they will be paid the minimum of the two reported values, and the traveler who reported a strictly lower value will receive dollars in reward from the other traveler. If you are one of the travelers, what will you report?
📗 Answer: .
[Note] Use the space to explain the steps or just take notes:
[L20Q]
Other students' answers:
In-class Quiz
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.
A \ B
I
II
III
I
II
III
IV
📗 Answer (comma separated vector): .
[Note] Use the space to explain the steps or just take notes:
📗 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
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. Use the checkboxes in the table to highlight the best responses: the first checkbox for the row player and the second checkbox for the column player.
A \ B
I
II
III
IV
I
II
III
IV
📗 Answer: .
[Note] Use the space to explain the steps or just take notes:
📗 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.
In-class Discussion
📗 Split or Steal games: (YouTube playlist: Link, Solution: 0).
📗 Sequential games can be converted into normal-form games where the actions are sequences of actions in the sequential games.
➩ The solution to the sequential game must be a Nash equilibrium of the normal-form game.
➩ Some Nash equilibria of the normal-form game may not be a solution (technically called a Subgame Perfect Equilibrium (SPE)) of the sequential game.
📗 The non-SPE Nash equilibria involve non-credible threats where the second player will use a non-optimal action in a subgame that causes the first player to use another action (so that the subgame will not be reached).
In-class Discussion
📗 [1 points] Country A can choose between Attack (A) and Not attack (N). If Country A chooses Not attack, the game ends, otherwise, Country B can choose to Fight (F) and Escape (E). Country A prefers (A, E) over (N) over (A, F). Country B prefers (N) over (A, E) over (A, F).
➩ How many Nash equilibria are there?
➩ What if Country B has the option to Burn the bridge (B) or Keep the bridge (K) at the beginning, and if B is chosen, the action E cannot be used?
📗 Answer (comma separated vector): .
[Note] Use the space to explain the steps or just take notes:
📗 Symmetric (simultaneous-move) games can have asymmetric Nash equilibria, but since the players are symmetric (identical), they might not know how to coordinate.
➩ Agreement before the game.
➩ Focal points (default actions).
➩ Randomized (mixed strategy).
In-class Discussion
📗 [1 points] There are three parking lots A, B, C with capacities \(15, 10, 5\). If a lot is below capacity, everyone in that lot gets \(+10\), and if a lot is above capacity, everyone in that lot gets \(-5\). Which parking lot would you choose?
A
B
C
[Note] Use the space to explain the reason you picked the action:
[L20Q]
Other students' answers:
In-class Discussion
📗 [1 points] There are three parking lots A, B, C with capacities \(15, 10, 5\). If a lot is below capacity, everyone in that lot gets \(+10\), and if a lot is above capacity, everyone in that lot gets \(-5\). Which parking lot would you choose?
A
B
C
[Note] Use the space to explain the reason you picked the action:
[L20Q]
Other students' answers:
In-class Discussion
📗 [1 points] There are three parking lots A, B, C with capacities \(15, 10, 5\). If a lot is below capacity, everyone in that lot gets \(+10\), and if a lot is above capacity, everyone in that lot gets \(-5\). Which parking lot would you choose?
A
B
C
[Note] Use the space to explain the reason you picked the action:
📗 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
\(-1\)
\(1\)
\(0\)
Nash Equilibrium:
[Note] Use the space to explain the steps or just take notes:
[L20Q]
In-class Quiz
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): .
[Note] Use the space to explain the steps or just take notes:
[L20Q]
In-class Quiz
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?
Romeo \ Juliet
Bach
Stravinsky
Bach
Stravinsky
📗 Answer: .
[Note] Use the space to explain the steps or just take notes:
📗 Best response dynamics starts with random action, and one player at a time use the best response to the other player's actions from the previous iteration.
➩ Time-averages converge to a mixed strategy Nash for zero-sum games.
➩ No result on convergence for general-sum games except for special classes of games such as potential games.
📗 Fictitious play algorithm uses best response to the mixed strategy to the other players' actions in the whole history.
In-class Quiz
ID:
📗 [1 points] Given the game , in step \(t\) = , the row player uses action , column player uses action , and fraction of times (including \(t\)) the row player uses each action is , and column player uses is , compute the next step of best response dynamics (row player first) and fictitious play in step \(t + 1\). The actions are \(\left\{0, 1, 2\right\}\) for both players.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
[Note] Use the space to explain the steps or just take notes:
[L20Q]
testtt,tdl,ie,zs,btb,pkg,pkt,pkr,cm,bos,brdq
📗 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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