Two-Stage Games

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Two-Stage Games. APEC 8205: Applied Game Theory Fall 2007. Objectives. Exercise Subgame Perfect Equilibrium on Some More Complicated Games . Two-Stage Games of Imperfect Information. The dynamic games we have played so far, have been ones of perfect information.

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Two-Stage Games

APEC 8205: Applied Game Theory

Fall 2007

Objectives
• Exercise Subgame Perfect Equilibrium on Some More Complicated Games
Two-Stage Games of Imperfect Information
• The dynamic games we have played so far, have been ones of perfect information.
• The games we want to look at know are dynamic games of imperfect information.
• For these games, the subgame perfect equilibrium will serve us well.
• Note that the two-stage game taxonomy is not particularly standard.
Application: Bank Run Game
• Who are the players?
• Two investors denoted by i = 1, 2.
• Who can do what when?
• Investors choose to withdraw savings (W) or not (N) in 1st stage.
• Investors choose to withdraw savings (W) or not (N) in 2nd stage.
• Who knows what when?
• Investors do not know each other’s choice in each stage.
• Stage 1 choices are reveal to each player before period 2 choices..
How are firms rewarded based on what they do?

1

W

N

2

Strategies?

W

N

W

N

{W, (N,W), (N,N)}

(r, r)

(D, 2r-D)

(2r-D, D)

1

W

N

2

W

N

W

N

Assumptions:

R >D > r > D/2

(R, R)

(2R-D, D)

(D, 2R- D)

(R, R)

Stage 2 Extensive Form Game:

1

W

N

2

W

N

W

N

(R, R)

(2R-D, D)

(D, 2R- D)

(R, R)

Subgame Perfect Equilibrium

Want to start by solving for Nash in stage 2?

Stage 2 Normal Form Game:

*

*

*

*

W is a dominant strategy for Player 1!

Assumptions:

R >D > r > D/2

W is a dominant strategy for Player 2!

(W, W) is a unique Nash equilibrium!

1

W

N

2

W

N

W

N

(r, r)

(D, 2r-D)

(2r-D, D)

(R, R)

Assumptions:

R >D > r > D/2

Subgame Perfect Equilibrium Continued

Lets use the Nash strategy (W, W) to rewrite the game and solve for Stage 1?

Revised Extensive Form Game

in Stage 1:

In Normal Form:

*

*

*

*

(W, W) is a Nash equilibrium!

So is (N, N)!

There is also a mixed strategy

Nash equilibrium!

Application: Tariffs & Imperfect International Competition
• Who are the players?
• Two countries denoted by i = 1, 2.
• Each country has a government.
• Each country has a firm where firms produce a homogeneous product.
• Who can do what when?
• First: Government in country i sets tariff (ti) on exports from firm in country j.
• Second: Firm in country i chooses how much to produce for domestic markets (hi) & how much to produce for export (ei)..
• Who knows what when?
• Governments do not know each others tariffs or firm outputs when choosing tariffs.
• Firms know tariffs, but not each other outputs when choosing outputs.
• Firm i’s reward includes
• Domestic Profit: (a – c – hi – ej)hi
• Export Profit: (a – c – tj – hj – ei)ei
• i(ti, tj, hi, ei, hj, ej) = (a – c – hi – ej)hi + (a – c – tj – hj – ei)ei
• Government i’s reward includes:
• Domestic Consumer Surplus: Qi2/2 where Qi = hi + ej
• Domestic Firm Profits: i(ti, tj, hi, ei, hj, ej)
• Tariff Revenues: tiej
• W(ti, tj, hi, ei, hj, ej) = Qi2/2 + i(ti, tj, hi, ei, hj, ej) + tiej
Subgame Perfect Equilibrium

Need to start by solving each firm’s optimal output decision.

First Order Conditions:

Now to the Government’s Optimization Problem:

First Order Condition:

Such that:

But what is the socially optimal tariff scheme?

First Order Conditions:

Such that:

Implications of Socially Optimal Policy
• Subsidize Exports
• Produce More for Export Markets & Less for Domestic Markets
• Total Output is Greater

What is going on here?

Application: Tournaments
• Who are the players?
• Two Workers & Boss
• Who can do what when?
• Boss determines how much to pay the most and least productive worker: wH & wL where wH > wL.
• Workers choose how hard to work: ei for i = 1, 2.
• Who knows what when?
• Boss knows output of each worker before making payment.
• Boss cannot observe effort perfectly due to random productivity shock: i with density f(i) & cumulative distribution F(i) for i = 1, 2.
• Assume E(i) = 0 for i = 1, 2 and independence of 1 & 2.
• Workers know pay schedule, but not the productivity shocks or other worker’s effort before choosing their own effort..
How are players rewarded based on what they do?
• Boss:
• yi(ei) = ei + i is ith workers output
• E(y1(e1) + y2(e2) – wH – wL) = e1 + e2 – wH – wL
• Worker i:
• wHPr(yi(ei) > yj(ej)) + wL(1 - Pr(yi(ei) > yj(ej))) – g(ei) for i≠ j
• Assume g’(ei) > 0 and g’’(ei) > 0.
Subgame Perfect Equilibrium

The last stage in this game is the workers’ choices of effort.

First Order Conditions:

For an Interior Solution:

Now what?

Note:

Bayes Rule Implies:

such that

Still, so what?

The workers are identical, so why not assume they will

choose the same equilibrium effort:

such that

A workers effort only depends on the difference in wages.

A useful result from this equation:

where w = wH – wL

Aside

Suppose i is normally distributed with variance 2.

Forging Ahead, We Now Turn to the Boss

Assume workers can work for someone else earning Ua.

For the boss to get these workers to work for him,

he must pay at least Ua on average:

wHPr(yi(ei) > yj(ej)) +wL (1 - Pr(yi(ei) > yj(ej))) – g(ei) ≥ Ua

But, if workers use the same effort in equilibrium:

(wH + wL)/2– g(ei) ≥ Ua

Optimization Problem for the Boss

subject to

(w + 2wL)/2– g(e*(w)) ≥ Ua

First Order Conditions:

For an interior solution, w > 0 &wL > 0 implies L/ w = 0 &L/ wL = 0,

such that  = 2, g’(e*(w)) = 1, and (w + 2wL)/2 – g(e*(w)) = Ua.

We are Almost There

Recall that

such that

How does this all really work out?

Suppose g(e) = ee where 1 >  > 0 and that i is normally distributed with variance 2:

g’(e*) = ee*= 1 implies e* = -ln()/

Implications
• Increasing the marginal cost of effort for a worker ()
• decreases equilibrium effort.
• increases the high and low equilibrium wage offered by the boss.
• does not affect the difference in equilibrium wages.
• Increasing a workers opportunity cost (Ua)
• does not affect equilibrium effort.
• increases the high and low equilibrium wage offered by the boss.
• does not affect the difference in equilibrium wages.
• Increasing the variability of output (2)
• does not affect equilibrium effort.
• decreases the low equilibrium wage offered by the boss.
• increases the high equilibrium wage offered by the boss.
• increases the difference in equilibrium wages.
Application: Rent Seeking with Endogenous Timing
• Who are the players?
• Two firms denoted by i = 1, 2 competing for a lucrative contract worth Vi.
• Who can do what when?
• Stage 1: firms cast ballots to choose who leads.
• Stage 2: firms choose effort (xi for i = 1, 2).
• Who knows what when?
• In 1st stage neither firm knows the other vote or effort.
• In 2nd stage, firms know each others 1st stage votes:
• If both vote for Firm i in 1st stage, Firm j sees Firm i’s effort before choosing.
• If both vote for different leader in 1st stage, a firm’s effort is chosen without knowing opponent’s effort..
• How are firms rewarded based on what they do?
• gi(xi,xj) = Vi xi / (xj + xj) – xi for i≠ j.
Subgame Perfect Equilibrium
• How many subgames are there?
• The whole game.
• Firm 1’s choice of effort, after Firm 2 when Firm 2 leads.
• Firm 2’s choice of effort, after Firm 1 when Firm 1 leads.
• Firm 1’s choice of effort, before Firm 2 when Firm 1 leads.
• Firm 2’s choice of effort, before Firm 1 when Firm 2 leads.
• Firm 1 and 2’s choice of effort when moving simultaneously.
• So there are lots of subgames, actually an infinite number.

We have actually seen the solution for all of these

subgames except the last one previously!

Here are Those Solutions
• i Leads & j Follows
• Strategies
• Rent Dissipation
• Payoffs
• Simultaneous Moves
• Strategies
• Rent Dissipation
• Payoffs
Lets Focus on the Solution to the Whole Game

Given the previous slide, the Normal form game is:

What is Firm i’s best response to Firm j voting for Firm i?

Firm i should vote for itself (Firm j) if

Firm i should prefer to vote for itself if Firm j votes for i!

What is Firm i’s best response to Firm j voting for itself?

Firm i should vote for itself (Firm j) if

Let i = Vi/Vj, which implies

or

Firm i should prefer to vote for itself if Firm j values winning more!

Firm i should prefer to vote for Firm j if Firm j values winning less!

Summary of Subgame Perfect Equilibrium
• If Vi > Vj
• both firms vote for Firm j to lead.
• Firm j chooses effort first:
• Firm i chooses effort second:
• Rent Dissipation is Vj/2
Implications
• Both Firms Agree About Who Should Go First
• Less Total Effort is Expended
• No Interventions Warranted
How did you do?

What is the subgame perfect Nash equilibrium?

How many subgames are there?
• Seven:
• (1) The Game As a Whole
• (2) Player 1’s Choice After Both Players Vote For Player 1 to Lead
• (6) Player 2’s Choice After Player 1
• (3) Player 2’s Choice After Both Players Vote For Player 2 to Lead
• (7) Player 1’s Choice After Player 2
• (4) Both Player’s Choices After Both Players Vote for Themselves to Lead
• (5) Both Player’s Choices After Both Players Vote for Their Opponent to Lead
(7) Player 1’s Choice After Player 2

*

*

*

If Player 2 chooses L, Player 1 should choose U.

If Player 2 chooses C, Player 1 should choose D.

If Player 2 chooses R, Player 1 should choose U.

*

*

*

*

Player 2 should choose C.

(6) Player 2’s Choice After Player 1

*

*

*

If Player 1 chooses U, Player 2 should choose L.

If Player 1 chooses M, Player 2 should choose R.

If Player 1 chooses D, Player 2 should choose R.

*

*

*

*

Player 1 should choose M.

(4) or (5) Both Player’s Choices After Both Players Vote for Themselves or Their Opponent to Lead

*

*

*

*

*

*

If Player 1 chooses U, Player 2 should choose L.

If Player 1 chooses M, Player 2 should choose R.

If Player 1 chooses D, Player 2 should choose R.

If Player 2 chooses L, Player 1 should choose U.

If Player 2 chooses C, Player 1 should choose D.

If Player 2 chooses R, Player 1 should choose U.

Therefore, (U, L) is the Nash Equilibrium!

(1) The Game As a Whole

*

*

*

*

If Player 1 votes for itself, Player 2 should vote for Player 1.

If Player 1 votes for 2, Player 2 should vote for itself.

If Player 2 votes for 1, Player 1 should vote for itself.

If Player 2 votes for itself, Player 1 should vote for itself.

The Nash Equilibrium is for both players to vote for Player 1 to Lead!

Player 1

Vote for Player 1 to Lead

If Follow,

Respond with U to L

Respond with D to C

Respond with U to R

If Simultaneous, Choose U

Player 2

Vote for Player 1 to Lead

If Follow,

Respond with L to U

Respond with R to M

Respond with R to D

If Simultaneous, Choose L

Summary of Subgame Perfect Equilibrium Strategies
Back to: How did you do?

Treatment 1

43%

14%

Simultaneous

43%

Only 1 out of 7 subgame perfect Nash!

Treatment 2

10%

30%

10%

10%

20%

20%

Only 1 out of 15 subgame perfect Nash equilibrium strategies submitted!