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Two-Stage Games

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Two-Stage Games

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)

Stage 2 Extensive Form Game:

1

W

N

2

W

N

W

N

(R, R)

(2R-D, D)

(D, 2R- D)

(R, R)

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

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:

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.

- wHPr(yi(ei) > yj(ej)) + wL(1 - Pr(yi(ei) > yj(ej))) – g(ei) for i≠ j

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

First Order Conditions:

For an Interior Solution:

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

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

g’(e*) = ee*= 1 implies e* = -ln()/

- 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!

Here are Those Solutions

- i Leads & j Follows
- Strategies
- Rent Dissipation
- Payoffs

- Simultaneous Moves
- Strategies
- Rent Dissipation
- Payoffs

Given the previous slide, the Normal form game is:

Firm i should vote for itself (Firm j) if

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

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

- 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

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

- (1) The Game As a Whole

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*

*

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.

*

*

*

*

*

*

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!

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*

*

*

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 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

Treatment 1

Disagreed on Leader

Agreed 2 Leads

43%

14%

Agreed 1 Leads

Agreed 2 Leads

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!