Prisoners' Dilemma

Cooperate

Defect

There's only one Nash equilibrium of this game.

Does it feel right? Do you live in a world in which nobody ever cooperates?

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The Economics of Ongoing Relationships

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 12

Today's Agenda

Part 1: Discrete Strategies

Part 2: Continuous Strategies

Finitely-repeated games

Evaluating infinite payoffs

Infinitely-repeated games

Collusion in Cournot Duopoly

Big Ideas

The ability to do so relies on
the ability to credibly promise/threaten something in the future,
and a high enough value placed on future payoffs

In ongoing relationships,
you can achieve things you cannot achieve in one-shot games.

What are we modeling?

Players who have ongoing relationships:
- Delta Airlines and American Airlines
- Managers and employees
- Spouses, roommates, friends
- Members of political parties
Decisions they make have an immediate impact but also an impact on the relationship.
How much greater is GDP because of the level of social capital (trust and other investments in relationships) is higher? How much worse off are we if that trust is eroded?

Finitely Repeated Games

Suppose there is some stage game G, which is repeated for T periods, with the payoffs being the sum of the payoffs in each period.
Trivial result: it is always a SPNE if a NE is played in every period.
Nontrivial result: there may be SPNE in which non-NE strategy profiles are played in early periods.
Key insight: this requires multiple Nash equilibria, which gives the ability to credibly promise a reward in the last period

If this game were played once,
what are the Nash equilibria?

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Recall: Strategy Definition

A strategy is a complete, contingent plan of action for a player in a game.

This means that every player
must specify what action to take
at every decision node in the game tree!

A strategy space is the set of all strategies available to a player.

What is player 2's strategy space?

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What is player 1's strategy space?

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What are the strategy spaces of the players if the game is played twice?

Remember: a strategy is a complete, contingent plan of how to play the game. It has to specify what to do after every possible history.

Each strategy must specify:

What to play in the first period.
What to play in the second period
for every possible outcome in the first period

Each strategy must specify:

What to play in the first period.
What to play in the second period
for every possible outcome in the first period

Remember: a strategy is a complete, contingent plan of how to play the game. It has to specify what to do after every possible history.

Each strategy must specify:

What to play in the first period.
What to play in the second period
for every possible outcome in the first period

Each strategy must specify:

What to play in the first period.
What to play in the second period
for every possible outcome in the first period

What are the strategy spaces of the players if the game is played twice?

Example strategy for player 1:

Play T in the first period.
In the second period, play T again after any history in which player 2 played L; after every other history, play M.

Note that this has to specify 10 things -- what to do initially, and what to do after each of the 9 histories of what could happen in the first period!

What are the subgames of this game?

Example strategy for player 1:

Play T in the first period.
In the second period, play T again after any history in which player 2 played L; after every other history, play M.

Note that this has to specify 10 things -- what to do initially, and what to do after each of the 9 histories of what could happen in the first period!

What are the subgames of this game?

Remember: in SPNE, the strategies must specify a NE in every subgame.

There were two NE of this game:
(T, L) and (M, C).

So, either (T, L) or (M, C) must be played in each of the 9 possible subgames in the second round in a subgame-perfect Nash equilibrium.

What are the subgames of this game?

Remember: in SPNE, the strategies must specify a NE in every subgame.

There were two NE of this game:
(T, L) and (M, C).

So, either (T, L) or (M, C) must be played in each of the 9 possible subgames in the second round in a subgame-perfect Nash equilibrium.

(T, L) or (M, C)

Let's look at the game again.

Both players would like to be able to play (B, R), but it wouldn't be possible in a one-shot game: player 1 would have an incentive to play T if they knew player 2 was going to play R.

But if it's played twice, can player 2 make any kind of credible promise to player 1 about how they'll act in the second round?

Consider player 2's strategy:

In the first round, play R.
If (B,R) was played in the first round, play L; otherwise play C.

Consider player 2's strategy:

In the first round, play R.
If (B,R) was played in the first round, play L; otherwise play C.

Assuming player 2 does this, what is player 1's best response in the second round if (B,R) was played in the first round?

What is player 1's best response in the second round if anything other than (B,R) was played in the first round?

Play T, since player 2 will play L.

Play M, since player 2 will play C.

Consider player 2's strategy:

In the first round, play R.
If (B,R) was played in the first round, play L; otherwise play C.

Player 1's best response:

In the first round, ___________________.
If (B,R) was played in the first round, play T; otherwise play M.

play B!

If player 1 plays B in the first period, they'll get 7, and then 5 in the second period when (T, L) is played, for a total of 12.

If player 1 plays T in the first period, they'll get 9, and then 2 in the second period when (T, L) is played, for a total of 11.

Infinitely Repeated Games

A stage game G is repeated an infinite number of times.
Can't just sum up the payoffs (would be infinite!)
Approach: use discounting, like we did in week 2

V(\pi_0,\pi_1,\pi_2,\pi_3, \cdots) = \pi_0 + \delta \pi_1 + \delta^2 \pi_2 + \delta^3 \pi_3 \cdots

V(x) = x + \delta x + \delta^2 x + \delta^3 x + \cdots

The present value of a stream of payoffs

(\(\pi_0\) now, \(\pi_1\) in the next period, \(\pi_2\) two periods from now, etc)

may be given by the infinite sum

The present value of a stream of payoffs of \(x\) in every period is:

V(x) = x + \delta [x + \delta x + \delta^2 x + \cdots]

Factor out \(\delta\) from the second and following terms:

What is this expression?

V(x) = x + \delta V(x)

(1-\delta)V(x) = x

V(x) = {x \over 1 - \delta}

The present value of a stream of payoffs of \(x\) in every period is:

\displaystyle{V(x,x,x,x, \cdots) = {x \over 1 - \delta}}

The present value of a stream of payoffs of \(z\) in every period,
starting next period, is:

\displaystyle{V(0,z,z,z, \cdots) = \delta \left[{z \over 1 - \delta}\right]}

The present value of getting
payoff \(y\) now, and then payoff \(z\) in every period thereafter, is:

\displaystyle{V(y,z,z,z, \cdots) = y + \delta \left[{z \over 1 - \delta}\right]}

Write these formulas down!

Strategies in Infinitely Repeated Games

Strategies must specify what to do in each period,
as a function of the history of all previously taken actions
We often group action histories to make strategy description simpler:

SPNE in Infinitely Repeated Games

Can't do backward induction!
Instead: "guess a strategy profile, and "check" that it is a Subgame Perfect Nash Equilibrium
A simple SPNE: play the Nash Equilibrium of the stage game in every period, for any history
Can we have a SPNE
in which there is cooperation?

‟Grim Trigger" Strategy (a.k.a. ‟Grudger")

Suppose your opponent is playing this strategy. Is playing this strategy a best response?

Cooperate

Defect

What is your payoff if you too play Grim Trigger?

What is your most profitable deviation?

Get a payoff of 2 in every period.

\displaystyle{V(5,5,5,5, \cdots) = {5 \over 1 - \delta}}

Get a payoff of 3 once, then 1 in every period.

\displaystyle{V(8,2,2,2, \cdots) = 8 + \delta \left[{2 \over 1 - \delta}\right]}

‟Grim Trigger" Strategy (a.k.a. ‟Grudger")

Suppose your opponent is playing this strategy. Is playing this strategy a best response?

What is your payoff if you too play Grim Trigger?

What is your most profitable deviation?

\displaystyle{V(5,5,5,5, \cdots) = {5 \over 1 - \delta}}

\displaystyle{V(8,2,2,2, \cdots) = 8 + \delta \left[{2 \over 1 - \delta}\right]}

When is GT a best response to GT?

\displaystyle{{5 \over 1 - \delta} > 8 + \delta \left[{2 \over 1 - \delta}\right]}

Multiply both sides by \(1 - \delta\):

5 > 8(1-\delta) + 2\delta

6\delta > 3

\delta > {1 \over 2}

If people are sufficiently patient (value future payoffs highly enough), then cooperation can be sustained using a "grim trigger" strategy.

Collusion

What is firm 2's best response function?

P(q_1,q_2) = 14 - (q_1+q_2)

c_2(q_2) = 2q_2

\pi_2(q_2) = P(q_1 + q_2)q_2 - c_2(q_2)

= (14 - q_1 - q_2)q_2 - 2q_2

= 14q_2 - q_1q_2 - q_2^2 - 2q_2

\pi'_2(q_2) = 14-q_1 - 2q_2 - 2 = 0

q_2^* = BR_2(q_1) = 6-\frac{1}{2}q_1

\text{total revenue}

\text{total cost}

\text{marginal revenue}

\text{marginal cost}

P(q_2|q_1) = 14-q_1 - q_2

MR(Q) = 14-q_1 - 2q_2

MC(q_2) = 2

6-\frac{1}{2}q_1

14-q_1

q_2

"Firm 2's Residual Demand Curve"

Firm 2's "best response function"

Cournot Best Response Functions

BR_1(q_2) = 6 - {1 \over 2}q_2

Firm 1's
best response function

BR_2(q_1) = 6 - {1 \over 2}q_1

Firm 2's
best response function

q_1

q_2

BR_1(q_2) = 6 - {1 \over 2}q_2

BR_2(q_1) = 6 - {1 \over 2}q_1

Nash equilibrium (a.k.a. Cournot equilibrium) is a pair of quantities \((q_1,q_2)\) such that

q_1 = BR_1(q_2)

q_2 = BR_2(q_1)

Profits in Cournot Equilibrium

In Cournot equilibrium, \(q_1 = q_2 = 4\).

\text{Combined profits = }\underbrace{P(q_1,q_2)q_1 - c(q_1)}_\text{Firm 1's profit} + \underbrace{P(q_1,q_2)q_2 - c(q_2)}_\text{Firm 2's profit}

= (14 - q_1 - q_2)q_1 - 2q_1 + (14 - q_1 - q_2)q_2 - 2q_2

Is this the best they could do? Suppose they each produced some amount \(q\).
What value of \(q\) would maximize their combined profits?

\text{Combined profits if they each produce 4 }= \underbrace{(14 - 4 - 4)4 - 2 \times 4}_{\pi_1(4,4) = 16} + \underbrace{(14 - 4 - 4)4 - 2\times 4}_{\pi_2(4,4) = 16}

\text{Combined profits if they each produce }q= \underbrace{(14 - q - q)q - 2q}_{\pi_1(q,q)} + \underbrace{(14 - q - q)q - 2q}_{\pi_2(q,q)}

\text{Combined profits if they each produce 4 }= \underbrace{(14 - 4 - 4)4 - 2 \times 4}_{\pi_1(4,4) = 16} + \underbrace{(14 - 4 - 4)4 - 2\times 4}_{\pi_2(4,4) = 16}

\text{Combined profits if they each produce }q= \underbrace{(14 - q - q)q - 2q}_{\pi_1(q,q)} + \underbrace{(14 - q - q)q - 2q}_{\pi_2(q,q)}

= 2[(14 - q - q)q - 2q]

= 2[14q - 2q^2 - 2q]

= 2[12q - 2q^2]

= 24q - 4q^2

Take derivative, set it equal to zero:

24 - 8q = 0

\Rightarrow q^* = 3

\text{Combined profits if they each produce 3 }= \underbrace{(14 - 3 - 3)3 - 2 \times 3}_{\pi_1(3,3) = 18} + \underbrace{(14 - 3 - 3)3 - 2\times 3}_{\pi_2(3,3) = 18}

\text{Combined profits if they each produce 4 }= \underbrace{(14 - 4 - 4)4 - 2 \times 4}_{\pi_1(4,4) = 16} + \underbrace{(14 - 4 - 4)4 - 2\times 4}_{\pi_2(4,4) = 16}

\text{Combined profits if they each produce 3 }= \underbrace{(14 - 3 - 3)3 - 2 \times 3}_{\pi_1(3,3) = 18} + \underbrace{(14 - 3 - 3)3 - 2\times 3}_{\pi_2(3,3) = 18}

But...if the other firm was going to produce 3, what should you produce?

BR_i(q_{-i}) = 6 - {1 \over 2}q_{-i}

BR_i(3) = 6 - {1 \over 2}3 = 4.5

\pi_i(4.5,3) = 20.25

COLLUSION

If both produce 3...

BETRAYAL

If you produce 4.5 while they produce 3...

NASH EQUILIBRIUM

If both produce 4

x = 18

y = 20.25

z = 16

\displaystyle{V(x,x,x,x, \cdots) = {x \over 1 - \delta}}

\displaystyle{V(y,z,z,z, \cdots) = y + \delta \left[{z \over 1 - \delta}\right]}

If both firms play a "grim trigger" strategy, when can they sustain collusion?

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COLLUSION

If both produce 3...

BETRAYAL

If you produce 4.5 while they produce 3...

NASH EQUILIBRIUM

If both produce 4

x = 18

y = 20.25

z = 16

\displaystyle{V(18,18,18, \cdots) = {18 \over 1 - \delta}}

\displaystyle{V(20.25,16,16, \cdots) = 20.25 + \delta \left[{16 \over 1 - \delta}\right]}

\displaystyle{{18 \over 1 - \delta} > 20.25 + \delta \left[{16 \over 1 - \delta}\right]}

18 > 20.25(1-\delta) + 16\delta

4.25\delta > 2.25

\delta > {2.25 \over 4.25} = {9 \over 17} \approx 0.53

What did we show this week?

Tuesday: some Nash equilibria are not subgame perfect, because they depend on non-credible threats or promises.
Today: some non-Nash outcomes of a stage game (e.g. cooperation in a Prisoners' Dilemma) can be sustained if the players have an ongoing relationship.
Sunday: homework for this week due
Next Monday: review session (time/place TBD, def not on zoom)
Next Wednesday: Game Theory Checkpoint I
(60 minutes long, in Hewlett 201 at either 5:30 or 7:00)