Econ 101 UCLA
Microeconomic Theory Spring 2026
Midterm 2 - Version A
Name: ID:
TA:
1
,Part 1: Short Questions (5 points each)
1. Consider a game of Rock-Paper-Scissors between Players 1 and 2. Winning yields a
payoff of 1, losing yields a payoff of −1, and drawing yields a payoff of 0. Which of the
following statements about this game is incorrect?
(a) Playing “Rock” can be Player 1’s best response to at least one strategy by Player
2.
(b) There exists a pure strategy Nash equilibrium.
(c) There exists a mixed strategy Nash equilibrium.
(d) There is no strictly dominant strategy for any player in this game.
(e) Each player has exactly three pure strategies and infinitely many mixed strategies
to choose from.
Solution: (b). There is no pure-strategy Nash Equilibrium in Rock-Paper-Scissors.
For any pure-strategy profile, the losing player always has a strict incentive to deviate
to the strategy that beats the opponent’s current choice (e.g., if Player 2 plays Rock,
Player 1 prefers Paper). Therefore, no pure-strategy profile can be a Nash Equilibrium,
making statement (b) false.
Statements (a), (c), (d), and (e) are all true. For (a): “Rock” is a best response
for Player 1 to any strategy by Player 2 that places sufficiently high probability on
“Scissors” (and is part of the best-response correspondence more generally). For (c):
the unique Nash Equilibrium of this game is in mixed strategies, in which each player
randomizes uniformly over the three actions, playing each with probability 13 . For (d):
no strategy is strictly dominant, since “Rock,” for example, beats “Scissors” but loses to
“Paper.” For (e): each player has exactly three pure strategies (Rock, Paper, Scissors)
and infinitely many mixed strategies, corresponding to all probability distributions
(pR , pP , pS ) with pR , pP , pS ≥ 0 and pR + pP + pS = 1.
2
, 2. Consider the sequential-move game represented by the game tree below. In the tree,
P1 refers to “Player 1” and P2 refers to “Player 2.”
P1
U D
P2 P2
L R L R
P1 P1 (4,2) (6,9)
A B A B
(5,8) (3,4) (2,6) (7,1)
Now consider the following four statements concerning this game:
• i. This game has a total of 5 subgames.
• ii. Player 2 has a total of 3 information sets in this game.
• iii. In the Subgame Perfect Nash Equilibrium of this game, Player 1’s payoff is 5.
• iv. In the Subgame Perfect Nash Equilibrium of this game, Player 2 plays L
regardless of what Player 1 does in the first move.
How many of the previous statements are true?
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
Solution: b). Only statement (i) is true. There are 5 subgames: the whole game; P2’s
decision node after U ; P1’s decision node after U -L; P1’s decision node after U -R; and
P2’s decision node after D. Statement (ii) is false: P2 has 2 information sets (one after
U and one after D), not 3. For statement (iii): by backward induction, after U -L, P1
plays A (payoff 5 > 3); after U -R, P1 plays B (payoff 7 > 2); hence after U , P2 chooses
L (getting 8 > 1). After D, P2 chooses R (getting 9 > 2). P1 then chooses D (payoff
6 > 5 from U ). P1’s equilibrium payoff is 6, not 5. Statement (iv) is false: P2 plays L
after U but R after D.
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Microeconomic Theory Spring 2026
Midterm 2 - Version A
Name: ID:
TA:
1
,Part 1: Short Questions (5 points each)
1. Consider a game of Rock-Paper-Scissors between Players 1 and 2. Winning yields a
payoff of 1, losing yields a payoff of −1, and drawing yields a payoff of 0. Which of the
following statements about this game is incorrect?
(a) Playing “Rock” can be Player 1’s best response to at least one strategy by Player
2.
(b) There exists a pure strategy Nash equilibrium.
(c) There exists a mixed strategy Nash equilibrium.
(d) There is no strictly dominant strategy for any player in this game.
(e) Each player has exactly three pure strategies and infinitely many mixed strategies
to choose from.
Solution: (b). There is no pure-strategy Nash Equilibrium in Rock-Paper-Scissors.
For any pure-strategy profile, the losing player always has a strict incentive to deviate
to the strategy that beats the opponent’s current choice (e.g., if Player 2 plays Rock,
Player 1 prefers Paper). Therefore, no pure-strategy profile can be a Nash Equilibrium,
making statement (b) false.
Statements (a), (c), (d), and (e) are all true. For (a): “Rock” is a best response
for Player 1 to any strategy by Player 2 that places sufficiently high probability on
“Scissors” (and is part of the best-response correspondence more generally). For (c):
the unique Nash Equilibrium of this game is in mixed strategies, in which each player
randomizes uniformly over the three actions, playing each with probability 13 . For (d):
no strategy is strictly dominant, since “Rock,” for example, beats “Scissors” but loses to
“Paper.” For (e): each player has exactly three pure strategies (Rock, Paper, Scissors)
and infinitely many mixed strategies, corresponding to all probability distributions
(pR , pP , pS ) with pR , pP , pS ≥ 0 and pR + pP + pS = 1.
2
, 2. Consider the sequential-move game represented by the game tree below. In the tree,
P1 refers to “Player 1” and P2 refers to “Player 2.”
P1
U D
P2 P2
L R L R
P1 P1 (4,2) (6,9)
A B A B
(5,8) (3,4) (2,6) (7,1)
Now consider the following four statements concerning this game:
• i. This game has a total of 5 subgames.
• ii. Player 2 has a total of 3 information sets in this game.
• iii. In the Subgame Perfect Nash Equilibrium of this game, Player 1’s payoff is 5.
• iv. In the Subgame Perfect Nash Equilibrium of this game, Player 2 plays L
regardless of what Player 1 does in the first move.
How many of the previous statements are true?
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
Solution: b). Only statement (i) is true. There are 5 subgames: the whole game; P2’s
decision node after U ; P1’s decision node after U -L; P1’s decision node after U -R; and
P2’s decision node after D. Statement (ii) is false: P2 has 2 information sets (one after
U and one after D), not 3. For statement (iii): by backward induction, after U -L, P1
plays A (payoff 5 > 3); after U -R, P1 plays B (payoff 7 > 2); hence after U , P2 chooses
L (getting 8 > 1). After D, P2 chooses R (getting 9 > 2). P1 then chooses D (payoff
6 > 5 from U ). P1’s equilibrium payoff is 6, not 5. Statement (iv) is false: P2 plays L
after U but R after D.
3
