346_lect4.pdf

Lecture Notes 4: Continuous Strategy Sets

Calculating Best Responses and Nash Equilibria

Suppose that player 1 chooses 𝑠𝑠₁ and player 2 chooses 𝑠𝑠₂. The payoffs to the players are as follows.

Π1 = 6𝑠𝑠1− 𝑠𝑠1𝑠𝑠2− 𝑠𝑠12

Π2 = 4𝑠𝑠2− 𝑠𝑠1𝑠𝑠2− 𝑠𝑠22

In the previous section, players chose strategies from discrete sets. The difference is that, in this example, players choose their strategies 𝑠𝑠₁ and 𝑠𝑠₂ from a continuous set. They can be any numbers.

Player 1 will choose 𝑠𝑠₁ in a way that maximizes his own payoff. Using calculus, the maximum can be characterized as the value of 𝑠𝑠₁ that sets the partial derivative of the payoff function Π₁ with respect to 𝑠𝑠₁ equal to zero.

𝜕𝜕Π1

𝜕𝜕𝑠𝑠1 = 0

6 − 𝑠𝑠2− 2𝑠𝑠1 = 0

𝑠𝑠1 = 3 −1_{2 𝑠𝑠}2

This is player 1’s best response function, just like in games with discrete strategies. It gives player 1’s best choice of 𝑠𝑠₁ depending upon the choice of strategy 𝑠𝑠₂ by player 2.

Similarly, player 2 will choose 𝑠𝑠₂ to maximize her own payoff Π₂.

𝜕𝜕Π2

𝜕𝜕𝑠𝑠2 = 0

4 − 𝑠𝑠1− 2𝑠𝑠2 = 0

𝑠𝑠2 = 2 −1_{2 𝑠𝑠}1

This is player 2’s best response function to any choice of 𝑠𝑠₁ by her opponent.

𝑠𝑠1 = 3 −1_{2 𝑠𝑠}2

𝑠𝑠1 = 3 −1_{2 �2 −}1_{2 𝑠𝑠}1�

𝑠𝑠1 =8₃

Substituting back into player 2’s best response function gives the following.

𝑠𝑠2 = 2 −1_{2 �}8_3�

= 2₃

Thus, the Nash Equilibrium strategies in this game are (𝑠𝑠₁, 𝑠𝑠₂) = �8 3,

2

3�. This is the only set of strategies where both players play are simultaneously playing a best response to each other.

2-Player Public Good Game

Consider a game where player 1 chooses 𝑥𝑥₁ and player 2 chooses 𝑥𝑥₂, and where the players have the following payoff functions.

Π1 = ln(𝑥𝑥1+ 𝑥𝑥2) + ln(24 − 𝑥𝑥1)

Π2 = ln(𝑥𝑥1+ 𝑥𝑥2) + ln(24 − 𝑥𝑥2)

An economic interpretation is that this is a public goods game. Provision of 𝑥𝑥₁ and 𝑥𝑥₂ benefits both players, but the player who provides it has to pay a cost to do so. For example, providing 𝑥𝑥₁ produces benefits for both players but imposes a cost on player 1.

Player 1’s best response is found by maximizing his payoff with respect to 𝑥𝑥₁.

𝜕𝜕Π1

𝜕𝜕𝑥𝑥1 = 0

1 𝑥𝑥1 + 𝑥𝑥2−

1

24 − 𝑥𝑥1 = 0

𝑥𝑥1 = 12 −1_{2 𝑥𝑥}2

𝜕𝜕Π2

𝜕𝜕𝑥𝑥2 = 0

1 𝑥𝑥1 + 𝑥𝑥2−

1

24 − 𝑥𝑥2 = 0

𝑥𝑥2 = 12 −1_{2 𝑥𝑥}1

The Nash Equilibrium strategies are where both players choose a best response. Substituting the second best response function into the first:

𝑥𝑥1 = 12 −1_{2 𝑥𝑥}2

𝑥𝑥1 = 12 −1_{2 �12 −}1_{2 𝑥𝑥}1�

𝑥𝑥1 = 8

Substituting back to player 2’s best response function gives 𝑥𝑥₂ = 8, so the Nash Equilibrium in this game is (𝑥𝑥₁, 𝑥𝑥₂) = (8,8).

To reiterate an important point that we made in unit 2.1, these are not the optimal strategy choices for maximizing total payoffs. In order to maximize the total payoff to the players, we would solve:

max Π1+ Π2 = 2ln(𝑥𝑥1+ 𝑥𝑥2) + ln(24 − 𝑥𝑥1) + ln(24 − 𝑥𝑥2)

Take the first-order conditions for this maximization problem:

𝜕𝜕Π1 + Π2

𝜕𝜕𝑥𝑥1 =

2 𝑥𝑥1+ 𝑥𝑥2−

1

24 − 𝑥𝑥1 = 0

𝜕𝜕Π1 + Π2

𝜕𝜕𝑥𝑥2 =

2 𝑥𝑥1+ 𝑥𝑥2−

1

24 − 𝑥𝑥2 = 0

Solving these equations gives 𝑥𝑥₁ = 12 and 𝑥𝑥₂ = 12. These strategy choices give both players a higher payoff than the Nash Equilibrium strategy set (𝑥𝑥₁, 𝑥𝑥₂) = (8,8). The problem is that they are not mutual best responses. For example, if player 2 chooses 𝑥𝑥₂ = 12 then player 1’s best

response is 𝑥𝑥₁ = 12 −1

2(12) = 6.

Symmetric Equilibria in n-player Games

Consider a version of the game above with 𝑛𝑛 players. Player 1 chooses 𝑥𝑥₁, player 2 chooses 𝑥𝑥₂, etc… The payoff functions are as follows.

Π1 = ln(𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛) + ln(24 − 𝑥𝑥1)

Π2 = ln(𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛) + ln(24 − 𝑥𝑥2)

⋮

Π𝑛𝑛 = ln(𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛) + ln(24 − 𝑥𝑥𝑛𝑛)

Solving this by hand would be very difficult. For example, if 𝑛𝑛 = 10, then you would have to take 10 derivatives and solve a system of 10 equations in 10 variables. However, for games like this, there is a “trick” that you can use to solve for the equilibrium without much difficulty.

Take the first-order condition to find the best response function for player 1:

𝜕𝜕Π1

𝜕𝜕𝑥𝑥1 =

1

𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛−

1

24 − 𝑥𝑥1 = 0

Notice now that this problem is perfectly symmetric across players. That is, if you simply change the player number, the payoff and best response functions have exactly the same structure for all players. This means that, in the solution, it must be the case that 𝑥𝑥₁ = 𝑥𝑥₂ = ⋯ = 𝑥𝑥_𝑛𝑛. With this in mind, we can rewrite the best response condition above.

1

𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥𝑛𝑛−

1

24 − 𝑥𝑥1 = 0

1

𝑥𝑥1+ 𝑥𝑥1+ ⋯ + 𝑥𝑥1−

1

24 − 𝑥𝑥1 = 0

1 𝑛𝑛𝑥𝑥1−

1

24 − 𝑥𝑥1 = 0

𝑥𝑥1 = _{1 + 𝑛𝑛}24

Because the problem is symmetric, we know that 𝑥𝑥₂ = 24 1+𝑛𝑛, 𝑥𝑥3 =

24

1+𝑛𝑛, etc… This technique for finding a Nash Equilibrium works only when the problem is symmetric. If payoff functions are not symmetric across players then it will not necessarily be true that 𝑥𝑥₁ = 𝑥𝑥₂ = ⋯ = 𝑥𝑥_𝑛𝑛 in equilibrium, and so this trick won’t work.

Tragedy of the Commons

Let’s see one more example of solving for a symmetric equilibrium in an n-player game.

A town has 99 shepherds. Each shepherd 𝑖𝑖 raises 𝑥𝑥_𝑖𝑖 sheep, and each sheep can be sold for $2000. The cost to a shepherd of raising a sheep is 300 + 𝑋𝑋, where 𝑋𝑋 = 𝑥𝑥₁+ ⋯ + 𝑥𝑥₉₉ represents the total number of sheep. The idea is that the sheep are grazed on public lands, but there is congestion – the higher the total number of sheep, the more costly it is to raise them.

Summarizing, the profit for shepherd 𝑖𝑖 is as follows.

Π𝑖𝑖 = 2000𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖(300 + 𝑋𝑋)

We went to determine how many sheep each shepherd will raise in the Nash Equilibrium. Write out the payoff function for the first shepherd.

Π1 = 2000𝑥𝑥1− 𝑥𝑥1(300 + 𝑥𝑥1+ 𝑥𝑥2+ ⋯ + 𝑥𝑥99)

= 2000𝑥𝑥1− 300𝑥𝑥1− 𝑥𝑥12− 𝑥𝑥1𝑥𝑥2− ⋯ − 𝑥𝑥1𝑥𝑥99

Shepherd 1 will choose the number of sheep 𝑥𝑥₁ to maximize his payoff.

𝜕𝜕Π1

𝜕𝜕𝑥𝑥1 = 2000 − 300 − 2𝑥𝑥1− 𝑥𝑥2− ⋯ − 𝑥𝑥99 = 0

But, because the problem is symmetric, it must be in equilibrium that 𝑥𝑥₁ = 𝑥𝑥₂ = ⋯ = 𝑥𝑥₉₉. Making the substitution into the first order condition above:

2000 − 300 − 2𝑥𝑥1− 𝑥𝑥1− ⋯ − 𝑥𝑥1 = 0

1700 − 2𝑥𝑥1− 98𝑥𝑥1 = 0

2𝑥𝑥1+ 98𝑥𝑥1 = 1700

100𝑥𝑥1 = 1700

𝑥𝑥1∗ = 17

Problems

1. Player 1 chooses 𝑥𝑥₁ and player 2 chooses 𝑥𝑥₂ in a simultaneous game with the following payoff functions.

Π1 = 100𝑥𝑥1− 𝑥𝑥12+ 20𝑥𝑥2

Π2 = 100𝑥𝑥2 − 𝑥𝑥22 + 20𝑥𝑥1𝑥𝑥2

a. Find the Nash Equilibrium.

b. Does either player have a dominant strategy? If so, what is it?

2. Michael and Nina are married. Michael invests effort level 𝑚𝑚 into the relationship and Nina invests effort level 𝑛𝑛. The payoff generated by the relationship is 12𝑚𝑚 + 12𝑛𝑛 + 𝑚𝑚𝑛𝑛. However, investing effort is costly. The cost for Michael to invest 𝑚𝑚 units of effort is given by 2𝑚𝑚2 and the cost for Nina to invest 𝑛𝑛 units of effort is 2𝑛𝑛2. Thus, the payoff functions are as given below. Find the Nash Equilibrium levels of effort.

Π𝑚𝑚 = 12𝑚𝑚 + 12𝑛𝑛 + 𝑚𝑚𝑛𝑛 − 2𝑚𝑚2

Π𝑛𝑛 = 12𝑚𝑚 + 12𝑛𝑛 + 𝑚𝑚𝑛𝑛 − 2𝑛𝑛2

3. There are 100 individuals, each individual 𝑖𝑖 donating 𝑥𝑥_𝑖𝑖 to a charity. Let 𝑋𝑋 indicate the total contribution over all individuals. The charity also has administrative expenses of 𝑃𝑃. Each individual cares about the total donations to the charity in excess of the charity’s administrative expenses. Specifically, each individual’s payoff is as follows.

Π𝑖𝑖 = 2√𝑋𝑋 − 𝑃𝑃 − 𝑥𝑥𝑖𝑖

a. What is each individual’s contribution to the charity in the symmetric Nash Equilibrium of this game?

4. A market consists of 10 firms that simultaneously choose their outputs. Each firm 𝑖𝑖’s output is given by 𝑥𝑥_𝑖𝑖. The price each firm can fetch on the market is given by the following.

𝑃𝑃1 = 1 − 𝑥𝑥1− 𝛽𝛽𝑥𝑥2 − 𝛽𝛽𝑥𝑥3− ⋯ − 𝛽𝛽𝑥𝑥10

𝑃𝑃2 = 1 − 𝑥𝑥2 − 𝛽𝛽𝑥𝑥1 − 𝛽𝛽𝑥𝑥3− ⋯ − 𝛽𝛽𝑥𝑥10

⋮

𝑃𝑃10= 1 − 𝑥𝑥10− 𝛽𝛽𝑥𝑥1− 𝛽𝛽𝑥𝑥3− ⋯ − 𝛽𝛽𝑥𝑥9

There are no costs, so each firm’s profit is just Π_𝑖𝑖 = 𝑥𝑥_𝑖𝑖𝑃𝑃_𝑖𝑖. For example:

Π1 = 𝑥𝑥1(1 − 𝑥𝑥1− 𝛽𝛽𝑥𝑥2− 𝛽𝛽𝑥𝑥3− ⋯ − 𝛽𝛽𝑥𝑥10)

a. How much output does each firm produce in the symmetric Nash Equilibrium? b. What is the market price in the symmetric Nash Equilibrium?