There are two players, 1 and 2. At the first stage, player 1 chooses whether to play the following simultaneous move game:

L R
U 2,2 0,0
D 0,0 4,4

If player 1 chooses "play" at the first stage, player 1 and 2 play the above simultaneous move game; If player 1 choose "not play" at the first stage, both players get the payoff of 3. How many SPE does the game have?

Question

There are two players, 1 and 2. At the first stage, player 1 chooses whether to play the following simultaneous move game:

L	R
U	2,2	0,0
D	0,0	4,4

If player 1 chooses "play" at the first stage, player 1 and 2 play the above simultaneous move game; If player 1 choose "not play" at the first stage, both players get the payoff of 3. How many SPE does the game have?

BlackTom AI · Accepted Answer

We start by restating the setup in our own terms to ensure the structure is clear. There is a first-stage decision for player 1: either to play the subsequent simultaneous-move game or not to play. If player 1 does not play, the outcome is a payoff of (3,3). If player 1 decides to play, the players then engage in the 2x2 stage game with payoffs:
- (U,L) → (2,2)
- (U,R) → (0,0)
- (D,L) → (0,0)
- (D,R) → (4,4)
This stage game itself has two pure Nash equilibria: (U,L) and (D,R). The first-stage decision is a strategic move anticipating the subgame outcomes.

Individual option analysis (given the typical SPNE framework):
- Option A: The continuation after playing yields (U,L) as the equilibrium outcome. In this case, if player 1 expects to obtain 2 by playing, he compares this to the immediate payoff of 3 from not playing. Since 2 < 3, player 1 would prefer not to play in the first stage. Therefore, this continuation cannot be part of a subgame perfect equilibrium, because it would not sustain the first-stage deviation incentives for player 1.

- Option B: The continuation after playing yields (D,R) as the equilibrium outcome. Here, playing leads to 4 for player 1, which is higher than the 3 he gets from not playing. Consequently, player 1 would prefer to play at the first stage, and the observed strategy profile can be sustained as a subgame perfect equilibrium provided the second-stage strategies implement (D,R) when the stage game is reached. In the subgame, (D,R) is a best-response pair for both players: given D, player 2’s best reply is R (4 vs 0); given R, player 1’s best reply to the continuation path is D (since the stage payoffs line up with the (D,R) outcome being a Nash equilibrium).

What this implies about SPNEs in the full game:
- There is a subgame perfect equilibrium in which player 1 chooses not to play in the first stage, yielding the immediate payoff (3,3).
- There is another subgame perfect equilibrium in which player 1 chooses to play in the first stage, and the continuation in the stage game resolves to the (D,R) Nash equilibrium, yielding (4,4).
Thus, there are two SPNE in this game.

Reasoning about why other potential continuations fail: if the stage game alongside the first stage could only resolve to (U,L), then player 1 would get 2 by playing versus 3 by not playing, disincentivizing the first-stage action to play; hence that path cannot be sustained in SPE. Since the (D,R) path can be sustained and yields a higher payoff for player 1 than not playing, it creates a second SPNE where playing occurs. Therefore, the count of SPNE is two.

In summary, the crucial points are: the stage game has two Nash equilibria; the first-stage choice by player 1 creates two potential SPNEs depending on which stage-game equilibrium is realized; the payoff comparison between playing (under the favorable equilibrium) and not playing yields two distinct equilibria, hence two SPNE in total.

23506 Strategic Decision Making and Game Theory - Autumn 2025 Online quiz 5

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