Lecture Notes 11: Games with Simultaneous and Sequential Moves
In this section, we will go through two elements of analyzing sequential games that require combining reasoning from simultaneous games and sequential games. First, we will show how to solve for subgame perfect equilibria in games that contain both simultaneous and sequential moves. Second, we will show how to formulate the “normal form” of a sequential game, which allows us to solve for the Nash Equilibria of a sequential game and contrast its Nash Equilibria with its subgame perfect equilibria.
Subgame Perfect Equilibrium Revisited
Earlier, we defined a subgame perfect equilibrium as the set of strategies found via backwards induction. This definition suffices for games of perfect information. But what about games that have both sequential and simultaneous moves embedded in them? A more technical definition of a subgame perfect equilibrium is that it requires playing a Nash Equilibrium in every subgame.
A subgame is an independent piece of the game. In games from the previous unit, saying that the SPE involved a Nash Equilibrium at every subgame was trivial. Each subgame consisted of a single move by one player. He chooses the move giving him the highest payoff, which is trivially a Nash Equilibrium choice. But what if a sequential game had a simultaneous game embedded in it? That is, rather than each subgame involving a singleton choice by one player, a subgame might involve simultaneous choices by multiple players.
Three Examples – Combining Simultaneous and Sequential Moves
Consider the game shown below. The game contains both sequential and simultaneous moves. Player 1 initially chooses 𝐴𝐴 or 𝐵𝐵. When player 1 chooses 𝐴𝐴, player 2 subsequently chooses between
Subgame perfect equilibrium requires a Nash Equilibrium at every subgame. The subgame comprising the simultaneous game has a single Nash Equilibrium at (𝐻𝐻, 𝐽𝐽) that leads to payoffs
(1,1). Now we can reason backwards from here.
• Player 2 chooses 𝑌𝑌 at her decision node.
• Player 1 is faced between choosing 𝐴𝐴 and getting a payoff of 2 (since player 2 follows by playing 𝑌𝑌) or choosing 𝐵𝐵 and getting a payoff of 1 (since choosing 𝐵𝐵 leads to the simultaneous game, where the Nash Equilibrium gives player 1 a payoff of 1). Given these choices, player 1 chooses 𝐴𝐴 at the initial node.
Altogether, the SPE of this game is (𝐴𝐴𝐻𝐻, 𝑌𝑌𝐽𝐽). Again, it is important to specify strategies at all nodes. Even though the outcome is that player 1 plays 𝐴𝐴 and player 2 plays 𝑌𝑌, we have to specify the strategies that would have been chosen in the simultaneous game, had it been reached.
The next example is somewhat more involved because there are multiple Nash Equilibria in the subgame comprising the simultaneous game. Consider the following game.
The simultaneous game actually has three Nash Equilibria.
• (𝐺𝐺, 𝐼𝐼) leading to payoffs of (6,4)
• (𝐻𝐻, 𝐽𝐽) leading to payoffs of (4,6)
• A mixed equilibrium �3 5𝐺𝐺 +
2 5𝐻𝐻,
2 5𝐼𝐼 +
3
5𝐽𝐽� leading to payoffs of (2.4,2.4)
• If (𝐺𝐺, 𝐼𝐼) is played in the simultaneous game, then player 1 chooses 𝐵𝐵 initially since his payoff from choosing 𝐴𝐴 is 3 but his payoff from choosing 𝐵𝐵 is 6. The equilibrium is
(𝐵𝐵𝐺𝐺, 𝑌𝑌𝐼𝐼).
• If (𝐻𝐻, 𝐽𝐽) is played in the simultaneous game, then player 1 chooses 𝐵𝐵 initially since his payoff from choosing 𝐴𝐴 is 3 but his payoff from choosing 𝐵𝐵 is 4. The equilibrium is
(𝐵𝐵𝐻𝐻, 𝑌𝑌𝐽𝐽).
• If the mixed equilibrium is played in the simultaneous game, then player 1 chooses 𝐴𝐴 initially since his payoff from choosing 𝐴𝐴 is 3 but his payoff from choosing 𝐵𝐵 and going to
the simultaneous game is 2.4. The equilibrium is ��𝐴𝐴,3 5𝐺𝐺 +
2
5𝐻𝐻� , �𝑌𝑌, 2 5𝐼𝐼 +
3 5𝐽𝐽��.
Here is one final example. Again, the idea is that a subgame perfect equilibrium can involve any Nash Equilibrium from the simultaneous-move subgame, which we then have to combine with best responses at the other nodes. We will restrict our attention to pure-strategy equilibria.
The subgame comprising the simultaneous game has three pure-strategy Nash Equilibria: (𝐿𝐿, 𝑍𝑍),
• If (𝐿𝐿, 𝑍𝑍) is used in the simultaneous game Player 2 chooses 𝐶𝐶 at her node (since 5 > 4)
Player 1 chooses 𝐵𝐵 at his node (since 5 > 3). The SPE is (𝐵𝐵𝐿𝐿, 𝐶𝐶𝑍𝑍). The payoffs from following this equilibrium path of play are (5,5).
• If (𝑀𝑀, 𝑋𝑋) is used in the simultaneous game Player 2 chooses 𝐷𝐷 at her node (since 6 >
5) Player 1 chooses 𝐴𝐴 at his node (since 3 > 2). The SPE is (𝐴𝐴𝑀𝑀, 𝐷𝐷𝑋𝑋). The payoffs from following this equilibrium path of play are (3,3).
• If (𝑁𝑁, 𝑌𝑌) is used in the simultaneous game Player 2 chooses 𝐷𝐷 at her node (since 7 > 5)
Player 1 chooses 𝐵𝐵 at his node (since 4 > 3). The SPE is (𝐵𝐵𝑁𝑁, 𝐷𝐷𝑌𝑌). The payoffs from following this equilibrium path of play are (4,7).
To summarize, this game has three pure-strategy SPE: (𝐵𝐵𝐿𝐿, 𝐶𝐶𝑍𝑍), (𝐴𝐴𝑀𝑀, 𝐷𝐷𝑋𝑋) and (𝐵𝐵𝑁𝑁, 𝐷𝐷𝑌𝑌).
Nash Equilibrium and Subgame Perfect Equilibrium
We will now move on to solving for the Nash Equilibria of a sequential game. In particular, we learn something deep about the notion of equilibrium in a sequential game by contrasting the Nash Equilibria and the subgame perfect equilibria of a sequential game.
To motivate this discussion, consider the sequential game below. The story is that there are two firms, each of which can choose to enter (𝐸𝐸) or not enter (𝑁𝑁) some market. Non-entrants always earn a profit of 0. A solo entrant earns monopoly profit of 100. But if both firms enter, the market is oversaturated and profit for each firm is −50.
To see why this is important, suppose that player 2 threatens to enter if player 1 enters. If player 1 actually believes this threat, then player 1’s best response is not to enter initially. This set of choices can be described by the strategy set (𝑁𝑁, 𝐸𝐸𝐸𝐸). With this strategy set, players are all making rational choices on the equilibrium path. Given that player 1 chooses 𝑁𝑁, player 2’s best response is 𝐸𝐸. And given that player 2’s strategy specifies entering at both nodes, player 1’s best response is 𝑁𝑁. However, this equilibrium is created by player 2 threatening to do something irrational off the equilibrium path. In other words, if player 1 actually did play 𝐸𝐸 (contrary to this strategy specification), it would not have been rational for player 2 to follow up by selecting 𝐸𝐸.
This example captures the essence of subgame perfection. (𝑁𝑁, 𝐸𝐸𝐸𝐸) is a Nash Equilibrium because the choices are rational on the equilibrium path. Players are free to make unreasonable threats at nodes that are not actually reached. But (𝑁𝑁, 𝐸𝐸𝐸𝐸) is not a subgame perfect equilibrium because SPE requires rational choices everywhere – both off the equilibrium path and on the equilibrium path.
We can be more systematic about computing Nash Equilibria of a sequential game. To do so, we need to write out the normal form of the sequential game. Player 1 has two strategies: 𝐸𝐸 and 𝑁𝑁. Player 2 has four strategies: 𝐸𝐸𝐸𝐸, 𝐸𝐸𝑁𝑁, 𝑁𝑁𝐸𝐸 and 𝑁𝑁𝑁𝑁. We enter these strategies into a game matrix and then fill in the payoffs by following the path provided by the strategies. For example, (𝐸𝐸, 𝐸𝐸𝐸𝐸) leads to the outcome with payoff vector (−50, −50). The strategy pair (𝐸𝐸, 𝑁𝑁𝐸𝐸) implies that player 2 chooses 𝑁𝑁 at her left node after player 1 chooses 𝐸𝐸, leading to payoff vector (100,0). The strategy pair (𝐸𝐸, 𝑁𝑁𝑁𝑁) also leads to the payoff vector (100,0). Since we stipulated that player 1’s strategy is 𝐸𝐸, player 2’s choice at the right node does not change the outcome.
Proceeding like this, the normal form of the sequential game above is as follows.
EE EN NE NN
E -50,-50 -50,-50 100,0 100,0
N 0,100 0,0 0,100 0,0
The pure-strategy Nash Equilibria of this game are {(𝐸𝐸, 𝑁𝑁𝐸𝐸), (𝐸𝐸, 𝑁𝑁𝑁𝑁), (𝑁𝑁, 𝐸𝐸𝐸𝐸)}. On the other hand, as we derived above, the only SPE of this game is (𝐸𝐸, 𝑁𝑁𝐸𝐸).
This is generally true. A subgame perfect equilibrium must be a Nash Equilibrium, but not the other way around. There can be Nash Equilibria that are not subgame perfect. Subgame perfect equilibrium is stricter than Nash Equilibrium because it requires rational choices at all subgames, whether they are actually reached or not. Nash Equilibrium requires rational choices only at nodes that are actually reached on the path of play.
Two More Examples – NE and SPE of a Sequential Game
Consider the sequential game below.
Standard backwards induction on the extensive form (game tree) gives that the SPE is (𝑈𝑈𝑈𝑈, 𝑋𝑋𝑌𝑌).
The normal form of the game is given below. Again, to insert the payoffs properly, just follow the path created by this strategy. For example, the strategy pair (𝑈𝑈𝐸𝐸, 𝑊𝑊𝑌𝑌) leads us first to 𝑈𝑈 and then to 𝑊𝑊, following player 2’s strategy. Thus, the corresponding payoff vector is (0,3). Other payoffs can be filled in similarly.
WY WZ XY XZ
UE 0,3 0,3 2,6 2,6
UF 0,3 0,3 6,4 6,4
DE 5,5 2,3 5,5 2,3
DF 5,5 2,3 5,5 2,3
The pure-strategy Nash Equilibria of the sequential game are the equilibria of the normal form given above. They are {(𝑈𝑈𝑈𝑈, 𝑋𝑋𝑌𝑌), (𝑈𝑈𝑈𝑈, 𝑋𝑋𝑍𝑍), (𝐷𝐷𝐸𝐸, 𝑊𝑊𝑌𝑌), (𝐷𝐷𝑈𝑈, 𝑊𝑊𝑌𝑌)}.
following 𝑈𝑈. But is that really credible? No, because player 2 is better off to select 𝑋𝑋 than to select
𝑊𝑊 if that node were reached. Thus, this cannot be a subgame perfect equilibrium because it does not involve rational choices at the nodes off of the equilibrium path.
Here is one final example.
Standard backwards induction gives that the SPE is (𝐶𝐶𝑋𝑋𝑋𝑋, 𝐿𝐿).
To find the game’s Nash Equilibria, we need to write out the normal form. Remember that a player’s strategy involves a choice at each node.
L R AWW 0,0 0,0
AWX 0,0 0,0
AXW 0,0 0,0
AXX 0,0 0,0
BWW 1,2 5,5
BWX 1,2 6,0
BXW 2,1 5,5
BXX 2,1 6,0
CWW 3,3 3,3
CWX 3,3 3,3
CXW 3,3 3,3
CXX 3,3 3,3
Problems
1. Consider the following game.
• Player 1 first chooses OUT or IN. If he chooses OUT, then the game is over with player 1 getting a payoff of 2 and player 2 getting a payoff of 0.
• If player 1 chooses IN, then player 2 chooses OUT or IN. If she chooses OUT, then the game is over with player 1 getting a payoff of 0 and player 2 getting a payoff of 2.
• If player 2 chooses IN, then both players play the following simultaneous game.
L R
U 3,1 0,-2
D -1,2 1,3
a. Draw the game tree.
b. Find all subgame perfect equilibria that involve pure strategies in the simultaneous game. What is the payoff in each equilibrium?
2. Consider the game below.
a. Find the subgame perfect equilibria that involve only pure strategies.
3. Find the subgame perfect equilibrium and all of the pure-strategy Nash Equilibria of the following game.
5. Consider the game below.
