Lecture Notes 2: Simultaneous Games – Discrete, Pure Strategies
Classifying Strategy Sets
Strategy sets can be discrete or continuous.
• A discrete strategy set means that a player chooses from a finite list of choices. For example, choosing whether to launch a missile or not launch a missile is a discrete choice.
• A continuous strategy set means that a player chooses from a continuum of choices. For example, choosing what price to set or what bid to submit in an auction are continuous strategy choices.
Players can either play pure strategies or mixed strategies.
• A player uses a pure strategy when he chooses a particular strategy with certainty.
• A player uses a mixed strategy when he randomizes over more than one strategy.
In this section, we will deal with discrete strategy sets and players who choose pure strategies. In later sections, we will deal with continuous strategy sets and with mixed strategies.
Best Responses
Two-player games with discrete strategy sets are typically represented as matrices. Player 1’s strategy choices are listed in rows and player 2’s strategy choices are listed in columns. In the game below, player 1 chooses from the strategy set {𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷} and player 2 chooses from the strategy set {𝑋𝑋, 𝑌𝑌, 𝑍𝑍}.
X Y Z
A 3,1 2,3 10,2
B 4,5 3,0 6,4
C 2,2 5,4 12,3
D 5,6 4,5 9,7
A best response for player 𝑖𝑖 is the strategy or strategies giving him the highest possible payoff, expressed as a function of the other player’s strategy choices. The key is that a player’s best response is contingent on the choices of the other players.
Consider the game above.
• If player 2 chooses 𝑋𝑋, then player 1’s best response is to play 𝐷𝐷. To see why, note that – when player 2 chooses 𝑋𝑋, player 1 gets a payoff of 3 by playing 𝐴𝐴, a payoff of 4 by playing
𝐵𝐵, a payoff of 2 by playing 𝐶𝐶 and a payoff of 5 by playing 𝐷𝐷. Thus, choosing 𝐷𝐷 gives player 1 the highest possible payoff when player 2 chooses 𝑋𝑋.
• If player 2 chooses 𝑌𝑌, then player 1’s best response is to play 𝐶𝐶 (since 5 is greater than 2, 3 or 4)
• If player 2 chooses 𝑍𝑍, then player 1’s best response is to play 𝐶𝐶 (since 12 is greater than 10, 6 or 9)
We could summarize these best responses with the following simple notation
𝐵𝐵𝐵𝐵1(𝑋𝑋) = 𝐷𝐷
𝐵𝐵𝐵𝐵1(𝑌𝑌) = 𝐶𝐶
𝐵𝐵𝐵𝐵1(𝑍𝑍) = 𝐶𝐶
The key point to emphasize is that player 1’s best responses depend upon player 2’s strategy choice. We can indicate these best responses by bolding them on the game matrix.
X Y Z
A 3,1 2,3 10,2
B 4,5 3,0 6,4
C 2,2 5,4 12,3
D 5,6 4,5 9,7
We now consider player 2’s best responses. These will be a function of player 1’s strategy choices.
• When player 1 plays 𝐵𝐵, player 2’s best response is 𝑋𝑋 (since 5 is greater than 0 or 4).
• When player 1 plays 𝐶𝐶, player 2’s best response is 𝑌𝑌 (since 4 is greater than 2 or 3).
• When player 1 plays 𝐷𝐷, player 2’s best response is 𝑍𝑍 (since 7 is greater than 6 or 5).
We can use notation to indicate these best responses:
𝐵𝐵𝐵𝐵2(𝐴𝐴) = 𝑌𝑌
𝐵𝐵𝐵𝐵2(𝐵𝐵) = 𝑋𝑋
𝐵𝐵𝐵𝐵2(𝐶𝐶) = 𝑌𝑌
𝐵𝐵𝐵𝐵2(𝐷𝐷) = 𝑍𝑍
Again, we indicate these best responses by bolding them on the game matrix.
X Y Z
A 3,1 2,3 10,2
B 4,5 3,0 6,4
C 2,2 5,4 12,3
D 5,6 4,5 9,7
Nash Equilibrium
A Nash Equilibrium is a set of strategies (one for each player) where all players are playing a best response to other players’ strategy choices. Put differently, at a Nash Equilibrium, no player can get a higher payoff by switching to a different strategy given the strategies chosen by other players in the equilibrium.
For the game in the previous section, (𝐶𝐶, 𝑌𝑌) is a Nash Equilibrium. This is the only set of strategies where both players are choosing a best response to the other player’s strategy choices. Given that player 2 plays 𝑌𝑌, then player 1 cannot get a higher payoff by switching to any strategy other than
𝐶𝐶 – Indeed, 𝐶𝐶 is the best response by player 1 to 𝑌𝑌. Similarly, given that player 1 plays 𝐶𝐶, then player 2 cannot get a higher payoff by playing something other than 𝑌𝑌. Both players are playing best responses to the other player’s strategy choices. This makes (𝐶𝐶, 𝑌𝑌) a Nash Equilibrium. Neither player can unilaterally switch strategies for a higher payoff.
Importantly, notice that a Nash Equilibrium does not necessarily give the highest payoff available in the game. For example, notice that (𝐷𝐷, 𝑍𝑍) gives both players a higher payoff than (𝐶𝐶, 𝑌𝑌). But
𝐶𝐶, not 𝐷𝐷. In other words, if the strategy pair were (𝐷𝐷, 𝑍𝑍), then player 1 can unilaterally deviate from 𝐷𝐷 to 𝐶𝐶 and will end up better off by doing so (since 12 > 9). This means that (𝐷𝐷, 𝑍𝑍) is not a Nash Equilibrium.
Here is another example.
L C R
T 1,1 3,0 1,0
M 0,3 1,1 2,4
B 0,1 4,2 3,3
Let us first find the best responses for player 1.
• When player 2 plays 𝐿𝐿, player 1’s best response is 𝑇𝑇
• When player 2 plays 𝐶𝐶, player 1’s best response is 𝐵𝐵
• When player 2 plays 𝐵𝐵, player 1’s best response is 𝐵𝐵
The best responses for player 2 are as follows.
• When player 1 plays 𝑇𝑇, player 2’s best response is 𝐿𝐿
• When player 1 plays 𝑀𝑀, player 2’s best response is 𝐵𝐵
• When player 1 plays 𝐵𝐵, player 2’s best response is 𝐵𝐵
Bolding these best responses on the game matrix, we see that (𝑇𝑇, 𝐿𝐿) and (𝐵𝐵, 𝐵𝐵) are both Nash Equilibria of this game.
L C R
T 1,1 3,0 1,0
M 0,3 1,1 2,4
B 0,1 4,2 3,3
Dominant Strategies
A strictly dominant strategy for player 𝑖𝑖 is a single strategy that is a strict best response to all possible strategies chosen by other players.
X Y Z
A 1,5 2,6 6,5
B 0,0 2,2 3,1
C 2,2 4,3 7,0
Player 1’s best response is 𝐶𝐶 regardless of player 2’s strategy. Whether player 2 chooses 𝑋𝑋, 𝑌𝑌 or
𝑍𝑍, the best response for player 1 is always to play 𝐶𝐶. Thus, 𝐶𝐶 is a strictly dominant strategy for player 1.
Similarly, player 2’s best response is 𝑌𝑌 regardless of what strategy player 1 chooses. Thus, 𝑌𝑌 is a strictly dominant strategy for player 2.
If a set of strategies comprises a strictly dominant strategy for every player in the game, then this set of strategies is called a dominant strategy equilibrium. In the game above, (𝐶𝐶, 𝑌𝑌) is a dominant strategy equilibrium since both 𝐶𝐶 and 𝑌𝑌 are strictly dominant strategies for their respective players.
A dominant strategy equilibrium is always a Nash Equilibrium, but not the other way around. In the first game from the previous section, (𝐶𝐶, 𝑌𝑌) is a Nash Equilibrium but 𝐶𝐶 is not a dominant strategy for player 1. In that case, 𝐷𝐷 is the best response when player 2 chooses 𝑋𝑋 – in other words,
𝐶𝐶 is not universally the optimal strategy choice for player 1. This means that (𝐶𝐶, 𝑌𝑌) is not a dominant strategy equilibrium. All strategies played by all players must be strictly dominant in a dominant strategy equilibrium.
A weakly dominant strategy for player 𝑖𝑖 is a strategy that is a weak best response to all possible strategies chosen by other players. Basically a weakly dominant strategy either gives a payoff that is strictly better or a tie. Consider the following game, again with best responses indicated.
L R
A 3,1 2,2
B 3,0 4,3
C 1,3 1,4
In this game, 𝐵𝐵 is a weakly dominant strategy for player 1. If player 2 plays 𝐿𝐿, then 𝐴𝐴 and 𝐵𝐵 are tied as best responses for player 1. When player 2 plays 𝐵𝐵, then 𝐵𝐵 is the unique best response by player 1. Thus, 𝐵𝐵 either gives the highest payoff or is a tie, for any strategy chosen by player 2.
(𝐵𝐵, 𝐵𝐵) is not a dominant strategy equilibrium in this game. Every strategy in a dominant strategy equilibrium must be strictly dominant for its respective player. But 𝐵𝐵 is only a weakly dominant strategy for player 1, not strictly dominant.
Dominated Strategies
A strictly dominated strategy for player 𝑖𝑖 is worse than some alternative strategy for all possible strategies chosen by other players. For example, consider the game below.
L C R
T 2,1 8,3 4,6
M 0,4 7,3 3,5
B 1,6 8,2 2,4
In this game, 𝑀𝑀 is strictly dominated by 𝑇𝑇 for player 1. Regardless of which strategy player 2 chooses, player 1 always gets a lower payoff by playing 𝑀𝑀 than he does by playing 𝑇𝑇.
Notice that 𝑀𝑀 is not strictly dominated by 𝐵𝐵. Although 𝐵𝐵 gives a better payoff for player 1 than 𝑀𝑀 if player 2 chooses 𝐿𝐿 or 𝐶𝐶, when player 2 chooses 𝐵𝐵, then 𝑀𝑀 gives a better payoff than 𝐵𝐵 for player 1. To be strictly dominated by another strategy, the dominated strategy has to be worse for all
possible strategies chosen by opponents.
A weakly dominated strategy for player 𝑖𝑖 is weakly worse than some alternative strategy for all possible strategies chosen by other players. Basically, weak domination allows for ties. Consider the game below.
X Y Z
A 4,2 1,3 1,2
B 3,4 0,4 1,3
C 2,1 0,2 0,0
For player 1, 𝐵𝐵 is weakly dominated by 𝐴𝐴. This is because 𝐵𝐵 either gives a lower payoff than 𝐴𝐴 (when player 2 chooses 𝑋𝑋 or 𝑌𝑌) or is a tie with 𝐴𝐴 (when player 2 chooses 𝑍𝑍). On the other hand, 𝐶𝐶 is strictly dominated by 𝐴𝐴 for player 1. We can also say that 𝐶𝐶 is weakly dominated by 𝐵𝐵.
Dominance, Domination and Nash Equilibrium
In this section, we will go through a few basic principles on the relationship between dominance, domination and Nash Equilibrium.
• If a player has a strictly dominant strategy, it will always be used in any Nash
Equilibrium of the game.
In fact, if all players have a strictly dominant strategy, then the Nash Equilibrium is a dominant strategy equilibrium.
• Strictly dominated strategies are never used in Nash Equilibrium. In fact, they can be eliminated iteratively.
By iterative elimination, we mean that once a strictly dominated strategy is eliminated, it might create another strictly dominated strategy, and that too can be eliminated. We demonstrate this procedure with the following game.
X Y Z
A 3,4 6,3 2,0
B 5,2 4,1 3,3
C 4,5 3,6 1,4
Strategy 𝐶𝐶 is strictly dominated by 𝐵𝐵 for player 1, so let us eliminate 𝐶𝐶.
X Y Z
A 3,4 6,3 2,0
B 5,2 4,1 3,3
C 4,5 3,6 1,4
In the game that is remaining, 𝑌𝑌 is strictly dominated by 𝑋𝑋 for player 2. Notice that this was not
true when strategy 𝐶𝐶 was present, but it is true with strategy 𝐶𝐶 eliminated. This is what we mean by iterated removal. We can now remove 𝑌𝑌.
X Y Z
A 3,4 6,3 2,0
B 5,2 4,1 3,3
In the game that is remaining, with 𝐶𝐶 and 𝑌𝑌 eliminated, strategy 𝐴𝐴 is now strictly dominated by 𝐵𝐵 for player 1. We can therefore remove 𝐴𝐴.
X Y Z
A 3,4 6,3 2,0
B 5,2 4,1 3,3
C 4,5 3,6 1,4
With only strategy 𝐵𝐵 remaining for player 1, 𝑋𝑋 is strictly dominated by 𝑍𝑍 for player 2, so we remove it.
X Y Z
A 3,4 6,3 2,0
B 5,2 4,1 3,3
C 4,5 3,6 1,4
This leaves (𝐵𝐵, 𝑍𝑍). In fact, if you solve this game using conventional best response analysis, you will find that (𝐵𝐵, 𝑍𝑍) is the only Nash Equilibrium. We say that this game is dominance-solvable, meaning that the Nash Equilibrium can be found by iterated removal of dominated strategies.
What about weakly dominated strategies – can we eliminate those as well? The general answer is no, for two reasons.
• Weakly dominated strategies can be a part of Nash Equilibria
For example, consider the game below, with best responses indicated.
L R
U 0,0 1,1
D 1,1 1,1
In this game, it is easy to see that 𝑈𝑈 is weakly dominated by 𝐷𝐷 for player 1 and that 𝐿𝐿 is weakly dominated by 𝐵𝐵 for player 2. However, we can see from the best response analysis that (𝑈𝑈, 𝐵𝐵),
(𝐷𝐷, 𝐿𝐿) and (𝐷𝐷, 𝐵𝐵) are all Nash Equilibria. This example suffices to illustrate that weakly dominated strategies can be used in Nash Equilibria.
• When eliminating weakly dominated strategies, a different order of removal can lead to different solutions
To illustrate this problem, consider the following game.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
Strategy 𝑈𝑈 is weakly dominated by 𝐷𝐷 for player 1, so let’s eliminate 𝑈𝑈.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
With 𝑈𝑈 eliminated, strategy 𝐿𝐿 is now weakly dominated by strategy 𝐵𝐵 for player 2, so let’s eliminate 𝐿𝐿.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
Now strategy 𝑀𝑀 is strictly dominated by 𝐷𝐷 for player 1, leaving (𝐷𝐷, 𝐵𝐵) as the solution obtained by iteratively removing weakly dominated strategies.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
With 𝑀𝑀 removed, strategy 𝐵𝐵 is weakly dominated by 𝐿𝐿 for player 2. Let’s eliminate it.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
With what is left, 𝑈𝑈 is strictly dominated by 𝐷𝐷 for player 1, leaving (𝐷𝐷, 𝐿𝐿) as the solution obtained by iteratively removing weakly dominated strategies in this order.
L R
U 5,1 4,0
M 6,0 3,1
D 6,4 4,4
The point is that a different order of removal can lead to different solutions if we try to solve a game by iteratively eliminating weakly dominated strategies. If we eliminate 𝑈𝑈 first, we get one solution. If we eliminate 𝑀𝑀 first, then we get a different solution. This is not an attractive feature. Iteratively removing weakly dominated strategies is not a reliable solution technique.
This problem does not exist for strictly dominated strategies. If a strategy is strictly worse than some other strategy for any choices by opponents, then order of removal is irrelevant. If a strategy is strictly dominated against all strategies by opponents, it will still be strictly dominated even if some of those strategies are removed.
Three-Player Games
Matrix X Matrix Y
L M L M
A 5,5,5 3,6,3 A 3,3,6 1,4,4
B 6,3,3 4,4,1 B 4,1,4 2,2,2
Best responses must be enumerated for all possible strategy choices by opponents. For example, here are player 1’s best responses:
• When player 2 chooses 𝐿𝐿 and player 3 chooses 𝑋𝑋, player 1’s best response is 𝐵𝐵.
• When player 2 chooses 𝑀𝑀 and player 3 chooses 𝑋𝑋, player 1’s best response is 𝐵𝐵.
• When player 2 chooses 𝐿𝐿 and player 3 chooses 𝑌𝑌, player 1’s best response is 𝐵𝐵.
• When player 2 chooses 𝑀𝑀 and player 3 chooses 𝑌𝑌, player 1’s best response is 𝐵𝐵.
Here are player 2’s best responses:
• When player 1 chooses 𝐴𝐴 and player 3 chooses 𝑋𝑋, player 2’s best response is 𝑀𝑀.
• When player 1 chooses 𝐵𝐵 and player 3 chooses 𝑋𝑋, player 2’s best response is 𝑀𝑀.
• When player 1 chooses 𝐴𝐴 and player 3 chooses 𝑌𝑌, player 2’s best response is 𝑀𝑀.
• When player 1 chooses 𝐵𝐵 and player 3 chooses 𝑌𝑌, player 2’s best response is 𝑀𝑀.
Here are player 3’s best responses:
• When player 1 chooses 𝐴𝐴 and player 2 chooses 𝐿𝐿, player 3’s best response is 𝑌𝑌.
• When player 1 chooses 𝐵𝐵 and player 2 chooses 𝐿𝐿, player 3’s best response is 𝑌𝑌.
• When player 1 chooses 𝐴𝐴 and player 2 chooses 𝑀𝑀, player 3’s best response is 𝑌𝑌.
• When player 1 chooses 𝐵𝐵 and player 2 chooses 𝑀𝑀, player 3’s best response is 𝑌𝑌.
We can shade these best responses on the game matrices.
Matrix X Matrix Y
L M L M
A 5,5,5 3,6,3 A 3,3,6 1,4,4
B 6,3,3 4,4,1 B 4,1,4 2,2,2
Let’s give one more example of a three-player game.
Matrix A Matrix B
L R L R
U 5,5,1 1,4,0 U 4,8,0 2,1,0
M 6,4,5 2,5,4 M 4,1,3 3,3,3
D 3,1,2 3,3,1 D 3,5,1 0,3,0
Player 1’s best responses are:
𝐵𝐵𝐵𝐵1(𝐿𝐿, 𝐴𝐴) = 𝑀𝑀
𝐵𝐵𝐵𝐵1(𝐿𝐿, 𝐵𝐵) = {𝑈𝑈, 𝑀𝑀}
𝐵𝐵𝐵𝐵1(𝐵𝐵, 𝐴𝐴) = 𝐷𝐷
𝐵𝐵𝐵𝐵1(𝐵𝐵, 𝐵𝐵) = 𝑀𝑀
Player 2’s best responses are:
𝐵𝐵𝐵𝐵2(𝑈𝑈, 𝐴𝐴) = 𝐿𝐿
𝐵𝐵𝐵𝐵2(𝑈𝑈, 𝐵𝐵) = 𝐿𝐿
𝐵𝐵𝐵𝐵2(𝑀𝑀, 𝐴𝐴) = 𝐵𝐵
𝐵𝐵𝐵𝐵2(𝑀𝑀, 𝐵𝐵) = 𝐵𝐵
𝐵𝐵𝐵𝐵2(𝐷𝐷, 𝐴𝐴) = 𝐵𝐵
𝐵𝐵𝐵𝐵2(𝐷𝐷, 𝐵𝐵) = 𝐿𝐿
Player 3’s best responses are:
𝐵𝐵𝐵𝐵3(𝑈𝑈, 𝐿𝐿) = 𝐴𝐴
𝐵𝐵𝐵𝐵3(𝑈𝑈, 𝐵𝐵) = {𝐴𝐴, 𝐵𝐵}
𝐵𝐵𝐵𝐵3(𝑀𝑀, 𝐿𝐿) = 𝐴𝐴
𝐵𝐵𝐵𝐵3(𝑀𝑀, 𝐵𝐵) = 𝐴𝐴
𝐵𝐵𝐵𝐵3(𝐷𝐷, 𝐿𝐿) = 𝐴𝐴
𝐵𝐵𝐵𝐵3(𝐷𝐷, 𝐵𝐵) = 𝐴𝐴
Problems
1. Find all Nash Equilibria of the game below.
L R
U 3,1 4,2
D 5,2 2,3
2. Find all Nash Equilibria of the game below.
Y Z
A 0,0 0,0
B 0,0 1,1
3. Find all Nash Equilibria of the game below.
L M R
U 2,9 5,5 6,2
S 6,4 9,2 5,3
D 4,3 2,7 7,1
4. Find all Nash Equilibria of the game below.
L C R
T 5,3 7,2 2,1
M 1,2 5,3 1,4
B 4,2 6,4 3,5
5. Solve the following game using iterated removal of strictly dominated strategies.
L M R
U 4,3 2,7 0,4
6. Consider the game below.
a. What are the Nash Equilibria?
b. Does player 1 have any strictly dominated strategies? If so, what are they? c. Does player 2 have any strictly dominated strategies? If so, what are they? d. Does player 1 have any weakly dominated strategies? If so, what are they? e. Does player 2 have any weakly dominated strategies? If so, what are they?
f. Which of the Nash Equilibria from (a) do not involve any weakly dominated strategies by either player?
X Y Z
A 0,0 0,0 0,0
B 0,0 1,1 2,0
C 0,0 0,2 2,2
