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# Two-Stage Games - PowerPoint PPT Presentation

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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APEC 8205: Applied Game Theory

Fall 2007

• Exercise Subgame Perfect Equilibrium on Some More Complicated Games

• 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.

• 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..

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)

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!

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!

• [W, W]

• [(N, W), (N, W)]

• 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

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

First Order Conditions:

For an Interior Solution do?These First Order Conditions Imply

Solving yields:

Such that:

First Order Condition:

Such that:

First Order Conditions:

Such that:

• Subsidize Exports

• Produce More for Export Markets & Less for Domestic Markets

• Total Output is Greater

What is going on here?

• 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..

• 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.

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

First Order Conditions:

For an Interior Solution:

Now what? do?

Note:

Bayes Rule Implies:

such that

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 do?

Suppose i is normally distributed with variance 2.

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

subject to

(w + 2wL)/2– g(e*(w)) ≥ Ua

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.

Recall that

such that

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 do?

• 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.

• 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!

• i Leads & j Follows

• Strategies

• Rent Dissipation

• Payoffs

• Simultaneous Moves

• Strategies

• Rent Dissipation

• Payoffs

Given the previous slide, the Normal form game is:

What is Firm do?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 do?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!

• 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 do?

• Both Firms Agree About Who Should Go First

• Less Total Effort is Expended

• No Interventions Warranted

What is the subgame perfect Nash equilibrium?

• 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

*

*

*

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.

*

*

*

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 for Themselves or Their Opponent to Lead

*

*

*

*

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 for Themselves or Their Opponent to Lead

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? for Themselves or Their Opponent to Lead

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!