two-stage games apec 8205: applied game theory fall 2007

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?

(R, R)

2

(2R-D, D) (D, 2R- D) (R, R)

1

W N W N

W N

(r, r)

2

(D, 2r-D) (2r-D, D)

1

W N WN

W N

Assumptions:

R > D > r > D/2

Strategies?

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

Subgame Perfect Equilibrium

Want to start by solving for Nash in stage 2?

(R, R)

2

(2R-D, D) (D, 2R- D) (R, R)

W N W N

W N

1

Stage 2 Extensive Form Game:

Player 2

W N

Player 1

W R

R D

2R - D

N 2R - D

D R

R

Stage 2 Normal Form Game:

Assumptions:

R > D > r > D/2

* **

*

W is a dominant strategy for Player 1!

W is a dominant strategy for Player 2!

(W, W) is a unique Nash equilibrium!

Subgame Perfect Equilibrium Continued

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

(R, R)

(r, r)

2

(D, 2r-D) (2r-D, D)

1

W N WN

W N

Assumptions:

R > D > r > D/2

Revised Extensive Form Game in Stage 1:

In Normal Form:

Player 2

W N

Player 1

W r

r 2r – D

D

N D

2r – D R

R

*

*

*

*

(W, W) is a Nash equilibrium!

So is (N, N)!

There is also a mixed strategyNash equilibrium!

Pure Strategy Subgame Perfect Equilibria Summarized

• [W, W]

• [(N, W), (N, W)]

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.

How are governments and firms rewarded based on what they do?

• 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: Qi

2/2 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


iijjijijjiijiieh

eehtcahehcaehehttii

,,,,,max0,0

02

jii

i ehcah

02

ijji

i ehtcae

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

0

ii

i hh

0ih

0

ii

i ee

0ie

First Order Conditions:

For an Interior SolutionThese First Order Conditions Imply

0*2*

0**2

0*2*

0**2

jii

ij

ijj

ji

ehtca

ehca

ehtca

ehca

3* i

i

tcah

3

2* j

i

tcae

Solving yields:

Such that:

9

2

9,

22ji

jii

tcatcatt

18

2

2

* 22ii tcaQ

3

2* ii

ji

tcatet

Now to the Government’s Optimization Problem:

3

2

9

2

918

2,max

222iijii

jiit

tcattcatcatcattW

i

0

3

4

9

22

9

2

iii

i

i tcatcatca

t

W

First Order Condition:

3*

cati

9

4*

cahi

9

*ca

ei

Such that:

But what is the socially optimal tariff scheme?

3

2

9

2

918

2

3

2

9

2

918

2,max

222

222

,

jjijj

iijiiji

tt

tcattcatcatca

tcattcatcatcattW

ji

0

9

84

3

4

9

2

9

2

iiii

i

tcatcatcatca

t

W


Such that:

0

9

84

3

4

9

2

9

2

jjjj

j

tcatcatcatca

t

W

2*

cati

6*

cahi

3

2*

caei

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.


0'Pr

ii

jjiiLH

i

i ege

eyeyww

e

g 0

ii

i ee

g

ii

jjiiLH eg

e

eyeyww '

Pr

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

ijjiiLjjiiHie

egeyeyweyeywgi

Pr1Prmax0


For an Interior Solution:

0ie

Now what?

jjjiji

jjii dfeefe

eyey

Pr

jijijjiijjii eeeeeyey PrPrPrNote:

such that

Bayes Rule Implies:

jjjijjjjijijiji dfeeFeeee 1Pr|PrPr

Still, so what?

jji

jjii dfe

eyey

2Pr

ijjLH egdfww '2

The workers are identical, so why not assume they will choose the same equilibrium effort:

such that

A useful result from this equation:

0

''

2

i

jji

eg

df

wd

de where w = wH – wL

A workers effort only depends on the difference in wages.

Aside

Suppose i is normally distributed with variance 2.

jjj dedfj

2

22 2

2

2

1

jde

j

2

2

22

2

22

1

22

2

2

1

jdej

2

2

22

1

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

LBww

wwwegL

2*2max0,0

subject to

(w + 2wL)/2 – g(e*(w)) ≥ Ua

a

LL Uweg

wwwwweL *

2

22*2


0

ww

L 0*'*'2

11*'2

wewegwew

L 0w

02

Lw

L0

LL

ww

L 0Lw

0*2

2

aL Uweg

wwL

0

L

0

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

ijjjjLH egdfwdfww '22

jj dfw

2

1

jj

aLdf

wegUw

22

1*

Recall that

jj

aHdf

wegUw

22

1*

such that

How does this all really work out?

2w

1aL Uw

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

1aH Uw

g’(e*) = ee* = 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.


• 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

otherwise

xVforxxV

V

V ijiij

j

i

0

0,

4

2

221

221

221

22

1 ,VV

VV

VV

VV

j

iij

j

i

V

VVV

V

V

4,

4

22

221

32

221

23

1 ,VV

V

VV

VV

2** i

ji

Vxx

21

2121 **

VV

VVxx

Lets Focus on the Solution to the Whole Game

Given the previous slide, the Normal form game is:

Player 2 Vote for 1 Vote for 2 Player 1

Vote for 1

2

21

12 4V

VVV

2

11 4V

VV

221

22

2VV

VV

221

21

1VV

VV

Vote for 2

221

22

2VV

VV

221

21

1VV

VV

1

22 4V

VV

1

22

21 4V

VVV

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

02 22 jjii VVVV

2

2

4 ji

ii

j

ii

VV

VV

V

VV

Firm i should vote for itself (Firm j) if

02 ji VV

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?

ij VV

2

22

2

2

11

4

111

1

ii

iii

i

ii VVV

i

jji

ji

ii V

VVV

VV

VV

4

2

2

2

Firm i should vote for itself (Firm j) if

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

Let i = Vi/Vj, which implies

i1 or

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

i

jj V

Vx

4*

2

otherwise

xVforxxVxx jijji

ji0

0*

Implications

• Both Firms Agree About Who Should Go First

• Less Total Effort is Expended

• No Interventions Warranted

How did you do?

Player 2 L: Nash C: Leader R: Follower

U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

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


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

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.

*

(3) Player 2’s Choice After Both Players Vote For Player 2 to Lead


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

Player 2 should choose C.

*

*

*

*

(6) Player 2’s Choice After Player 1


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

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.

*

(2) Player 1’s Choice After Both Players Vote For Player 1 to Lead


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

Player 1 should choose M.

*

*

**

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


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

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

Player 2 Player 1 Leads Player 2 Leads Player 1

Player 1 Leads

34 32

27 24

Player 2 Leads

27 24

30 15

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!

Summary of Subgame Perfect Equilibrium Strategies

• Player 1– Vote for Player 1 to Lead

– If Lead, Choose M.

– 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 Lead, Choose C

– If Follow,• Respond with L to U

• Respond with R to M

• Respond with R to D

– If Simultaneous, Choose L

Back to: How did you do?


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

43%

43%

14%

Treatment 1


U: Nash 27

24 24

10 26

34 Player 1

M: Leader 33

18 29

12 34

32

D: Follower 31

20 30

15 32

28

10%

20%

30%

Treatment 2

Agreed 2 Leads

Disagreed on Leader

Agreed 1 LeadsAgreed 2 LeadsSimultaneous

Only 1 out of 7 subgame perfect Nash!

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

20%

10%

10%

two-stage games apec 8205: applied game theory fall 2007

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