Backwards Induction. Extensive-Form Representation. Backwards Induction (cont ) The player 2 s optimization problem in the second stage

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1 Lecture Notes II- Dynamic Games of Complete Information Extensive Form Representation (Game tree Subgame Perfect Nash Equilibrium Repeated Games Trigger Strategy Dynamic Games of Complete Information Dynamic game with complete information Sequential games in wch the players payoff functions are common knowledge Perfect (imperfect information: For each move in the play of the game, the player with the move knows (doesn t know the full story of the play of the game so far Dynamic Game of Complete and Perfect Information Key features ( the moves occur in sequence ( all previous moves are observed before the next move is chosen ( the players playoff from each feasible combination of moves are common knowledge Backwards Induction A simple dynamic game of complete and perfect information. Player chooses an action a from the feasible set A. Player observes a and then chooses an action a from the feasible set A Payoffs are u (a,a and u (a,a Backwards Induction (cont The player s optimization problem in the second stage max u( a, a a A Assume that for each a in A, players optimization problem has a unique solution, denoted by R (a. Ts player s reaction (or best response to player s action The player s optimization problem in the first stage max u ( a, R ( a a A Assume that ts optimization problem for player also has a unique solution denoted by a * We call (a *,R *(a * the backwards induction outcome of ts game Extensive-Form Representation L L R R L In the first stage, player play the optimal action L R

2 Example : Stackelberg Model of Duopoly Timing of the game ( firm chooses a quantity q ( firm observes q then choose a quantity q Demand function P(Qa-Q, Qq +q Profit function to firm i (q i,q j q i [P(Q-c] Example : Stackelberg Model of Duopoly (cont In the second stage, firm s reaction to an arbitrary quantity by firm R (q is given by solving max ( q, q max q [ a q q c] a q c R ( q In the first stage, firm s problem is to solve max ( q, R ( q max q [ a q R ( q c] max q Outcome q q a q c * a c q * a c a c q and R ( q* 4 q q q Example : Stackelberg Model of Duopoly (cont Compare with Nash equilibrium of the simultaneous Cournot game Decide simultaneously a c ( a c a + c q* q*, Q*, p Decide sequentially a c a c ( a c a + c q*, R ( q*, Q*, p Example : Stackelberg Model of Duopoly (cont Decide simultaneously a c a c ( a c * * 9 Decide sequentially ( ( * a c a c a c, * a c a c a c In single-person decision theory, having more information can never make the decision worse off, In game theory, however, having more information can make a player worse off Two-Stage Game of Complete but Imperfect Information A two-stage game Players and simultaneously choose actions a and a from feasible sets A and A, respectively Players and 4 observe the outcome of the first stage, (a,a, and then simultaneously choose actions a and a 4 from feasible sets A and A 4, respectively Payoffs are u i (a,a,a,a 4 for i,,,4 Two-Stage Game of Complete but Imperfect Information (cont Backward induction For any feasible outcome of the first-stage game, (a,a, the second stage that remains between players and 4 has a unique Nash equilibrium (a *(a,a, a 4 *(a,a Subgame-perfect outcome Suppose (a *,a * is the unique Nash equilibrium of simultaneous-move game of player and player (a *,a *,a *(a *,a *,a 4 *(a *,a * is called subgame-perfect outcome

3 Example : Tariffs and Imperfect International Competition Two identical countries, denoted by i, Each country has a government that chooses a tariff rate t i, a firm that produces output for both home consumption h i and export e i If the total quantity on the market in country i is Q i, then the market-clearing price is P i (Q i a-q i, where Q i h i +e j The total cost of production for firm i is C i (h i,e i c(h i +e i Example : Tariffs and Imperfect International Competition (cont Timing of the game First, the governments simultaneously choose tariff rates t and t Second, the firms observe the tariff rates and simultaneously choose quantities for home consumption and for export (h,e and (h,e Payoffs are profit to firm i and total welfare to government i welfare consumers surplus + firms profit +tariff revenue Example : Tariffs and Imperfect International Competition (cont Firm i s profit i( ti, t j,, ei, hj, e j [ a ( + e j ] + [ a ( ei + hj ] ei c( + ei t jei Government i s payoff a a-q Wi ( ti, t j,, ei, hj, e j Qi + i( ti, t j,, ei, hj, e j + tie j P Pa-Q Consumers surplus Example : Tariffs and Imperfect International Competition (cont Firm i s optimization problem * * max i( ti, t j,, ei, hj, e j, ei max [ a ( + e j* c] * max ei[ a ( hj + ei c] t jei ei a c + ti * * ( * hj a e j c ei * ( a hj * c t j a c t j ei * Q i Q Example : Tariffs and Imperfect International Competition (cont Government i s optimization problem (( a c t ( ( * i a c + t a c t i j ti( a c ti max Wi *( ti, t j* ti a c a c + t 4( ti * * i a c a c t j a c ei * 9 9 5( a c Qi + e j 9 Implication t i * ( a c Qi (Cournot s model, gher consumers surplus max W *( t, t + W *( t, t t, t t* t* (free trade i j Player Two-Stage Repeated Game L R Player L R, 5,,5 4,4 L Player R Prisoner L R, 6,,6 5,5 The unique subgame-perfect outcome of the two-stage Prisoners Dilemma is (L,L in the first stage, followed by (L,L in the second stage Cooperation, that is, (R,R cannon be aceved in either stage of the subgame-perfect outcome

4 Finitely Repeated Game Definition Given a stage game G, let G(T denote the finitely repeated game in wch G is played T times, with the outcomes of all proceeding plays observed before the next play begins. The playoffs for G(T are simply the sum of the playoffs from the T stage games Proposition If the stage game G has a unique Nash equilibrium then, for any finite T, the repeated game G(T has a unique subgame-perfect outcome: the Nash equilibrium of G is played in every stage Finitely Repeated Game with Multiple Nash Equilibrium (cont Cooperation can be aceved in the first stage of a subgame-perfect outcome of the repeated game If G is a static game of complete information with multiple Nash equilibria then in wch, for any t<t, there may be subgame-perfect outcome in stage t is not a Nash equilibrium of G Implication: credible threats or promises about future behavior can influence current behavior Infinitely Repeated Games Present value : Given the discount factor δ, the present value of the infinite sequence of payoffs,,,,... is t + δ + δ + L δ t t Trigger strategy: player i cooperates until someone fails to cooperate, wch triggers a switch to noncooperation forever after Trigger strategy is Subgame perfect Nash equilibrium when δ is sufficiently large Implication: even if the stage game has a unique Nash equilibrium, there may be subgame-perfect outcomes of the infinitely repeated game in wch no stage s outcome is a Nash equilibrium Infinitely Repeated Games: Example Player L R Player L R, 5,,5 4,4 Trigger strategy Play R i in the first stage. In the t th stage, if the outcome of all t- proceeding stages has been (R,R then play R i ; otherwise, play L i Infinitely Repeated Games in Example (cont Player L R Player L R, 5,,5 4,4 If any player deviates δ V d 5 + δ + δ + L 5 + δ If no player deviates 4 V c ( Solve V 4 + δv δ Condition for both players to play the trigger strategy (Nash equilibrium 4 δ Vc Vd 5 δ δ + δ 4 Infinitely Repeated Games in Example (cont Player L R Player L R, 5,,5 4,4 The trigger strategy is a subgame perfect Nash equilibrium (Proof The infinitely repeated game can be grouped into two classes: ( Subgame in wch all the outcomes of earlier stages have been (R,R Again the trigger strategy, wch is Nash equilibrium of the whole game ( Subgames in wch the outcome of at least one earlier stage differs from (R,R Repeat the stage-game equilibrium (L,L,wch is also Nash equilibrium of the whole game 4

5 Example : Collusion between Cournot Duopolists Trigger strategy Produce half monopoly quantity,q m /, in the first period. In the tth period, produce q m / if both firms have produced q m / in each of the t- previous periods; otherwise, produce the Cournot quantity Example : Collusion between Cournot Duopolists (cont Collusion profit Deviation profit m ( a c 8 9( a c d 64 Competition profit ( a c C 9 d max a qd qm c qd qd ( a c Solve FOC qd 8 5( a c Q 8 ( a c ( a c d 8 8 Condition for both producer to play trigger strategy δ m d C δ + δ 9 δ 7 Example 4: Efficiency Wages The firms induce workers to work hard by paying gh wages and threatening to fire workers caught srking (Shapiro and Stiglitz 984 Stage game First, the firms offers the worker a wage w Second, the worker accepts or rejects the firm s offer If the worker rejects w, then the worker becomes selfemployed at wage w If the worker accepts w, then the worker chooses either to supply effort (wch entails disutility e or to srk (wch entails no disutility Example 4: Efficiency Wages (cont The worker s effort decision is not observed by the firm, but the worker s output is observed by both the firm and the worker Output can be either gh (y or low ( If the worker supplies effort then output is sure to be gh If the worker srks then output is gh with probability p and low with probability -p Low output is an incontrovertible sigh of srking Payoffs: Suppose the firm employs the worker at wage w if the worker supplies effort and output is gh, the playoff of the firm is y-w and playoff of the worker is w-e Efficient employment y-e>w >py Example 4: Efficiency Wages (cont Subgame-perfect outcome The firm offer w and the worker chooses selfemployment The firms pays in advance, the worker has no incentive to supply effort Trigger strategy as repeated-game incentives The firm s strategy: offer ww* (w*>w in the first period, and in each subsequent period to offer ww* provided that the story of play is gh-wage, ghoutput, but to offer w, otherwise The worker s strategy: accept the firm s offer if w>w (choosing self-employment otherwise the story of play, is gh-wage, gh-output (srking otherwise Example 4: Efficiency Wages (cont If it is optimal for the worker to supply effort, then the present value of the worker s payoff is V ( w* e + δ ( w* e /( δ e V e If it is optimal for the worker to srk, then the (expected present value of the worker s payoffs is w ( δ w* + δ ( p w Vs w* + δ pvs + ( p V s δ ( δp( δ V e 5

6 Example 4: Efficiency Wages (cont It is optimal for the worker to supply effort if Ve V s ( pδ e δ w* w + w + + e δ ( p δ ( p The firm s strategy is a best response to the worker s if y > w* We assume y e > w, the SPNE implies ( δ e y e w + δ ( p Homework # Problem set 6,, 8,,5,7 (from Gibbons Due date two weeks from current class meeting Bonus credit Propose new applications in the context of IT/IS or potential extensions from examples discussed 6