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 1 st stage. –Investors choose to withdraw savings (W) or not (N) in 2 nd 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 WNWN WN (r, r) 2 (D, 2r-D)(2r-D, D) 1 WNW N WN 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) WNWN WN 1 Stage 2 Extensive Form Game: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 WNW N WN Assumptions: R > D > r > D/2 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!

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 (t i ) on exports from firm in country j. –Second: Firm in country i chooses how much to produce for domestic markets (h i ) & how much to produce for export (e i ).. 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 – h i – e j )h i –Export Profit: (a – c – t j – h j – e i )e i –  i (t i, t j, h i, e i, h j, e j ) = (a – c – h i – e j )h i + (a – c – t j – h j – e i )e i Government i’s reward includes: –Domestic Consumer Surplus: Q i 2 /2 where Q i = h i + e j –Domestic Firm Profits:  i (t i, t j, h i, e i, h j, e j ) –Tariff Revenues: t i e j –W(t i, t j, h i, e i, h j, e j ) = Q i 2 /2 +  i (t i, t j, h i, e i, h j, e j ) + t i e j

Subgame Perfect Equilibrium Need to start by solving each firm’s optimal output decision. First Order Conditions:

For an Interior Solution These First Order Conditions Imply Solving yields: Such that:

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: w H & w L where w H > w L. –Workers choose how hard to work: e i 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: –y i (e i ) = e i +  i is ith workers output –E(y 1 (e 1 ) + y 2 (e 2 ) – w H – w L ) = e 1 + e 2 – w H – w L Worker i: –w H Pr(y i (e i ) > y j (e j )) + w L (1 - Pr(y i (e i ) > y j (e j ))) – g(e i ) for i ≠ j Assume g’(e i ) > 0 and g’’(e i ) > 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: such that Bayes Rule Implies:

Still, so what? The workers are identical, so why not assume they will choose the same equilibrium effort: such that A useful result from this equation: where  w = w H – w L A workers effort only depends on the difference in wages.

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 U a. For the boss to get these workers to work for him, he must pay at least U a on average: w H Pr(y i (e i ) > y j (e j )) +w L (1 - Pr(y i (e i ) > y j (e j ))) – g(e i ) ≥ U a But, if workers use the same effort in equilibrium: (w H + w L )/2 – g(e i ) ≥ U a

Optimization Problem for the Boss subject to (  w + 2w L )/2 – g(e*(  w)) ≥ U a

First Order Conditions: For an interior solution,  w > 0 & w L > 0 implies  L/  w = 0 &  L/  w L = 0, such that = 2, g’(e*(  w)) = 1, and (  w + 2w L )/2 – g(e*(  w)) = U a.

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 (U a ) –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 V i. Who can do what when? –Stage 1: firms cast ballots to choose who leads. –Stage 2: firms choose effort (x i for i = 1, 2). Who knows what when? –In 1 st stage neither firm knows the other vote or effort. –In 2 nd stage, firms know each others 1 st stage votes: If both vote for Firm i in 1 st stage, Firm j sees Firm i’s effort before choosing. If both vote for different leader in 1 st stage, a firm’s effort is chosen without knowing opponent’s effort.. How are firms rewarded based on what they do? –g i (x i, x j ) = V i x i / (x j + x j ) – x i 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 Firm i should prefer to vote for itself if Firm j values winning more! Let  i = V i /V j, which implies or Firm i should prefer to vote for Firm j if Firm j values winning less!

Summary of Subgame Perfect Equilibrium If V i > V j –both firms vote for Firm j to lead. –Firm j chooses effort first: –Firm i chooses effort second: –Rent Dissipation is V j /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. *

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

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

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? 43% 14% Treatment 1 10% 20% 30% Treatment 2 Agreed 2 Leads Disagreed on Leader Agreed 1 Leads Agreed 2 Leads Simultaneous Only 1 out of 7 subgame perfect Nash! Only 1 out of 15 subgame perfect Nash equilibrium strategies submitted! 20% 10%