Introduction to Game Theory

Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 1. Dynamic games of complete and perfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 1 / 47

Topics covered 1 Theory: Backwards induction 2 Stackelberg model of duopoly 3 Wages and employment in a unionized firm 4 Sequential bargaining V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 2 / 47

Important words We introduce dynamic games We restrict our attention to games with complete information The players payoff functions are common knowledge In this chapter we analyze dynamic games with complete but also perfect information At each move in the game, the player with the move knows the full history of the play of the game thus far The central issue in dynamic games is credibility V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 3 / 47

An example Consider the following 2-move game First, player i 1 chooses between giving player i 2 $1,000 and giving player i 2 nothing Second, player i 2 observes player i 1 s move and then chooses whether or not to explode a grenade that will kill both players Suppose that player i 2 threatens to explode the grenade unless player i 1 pays the $1,000 If player i 1 believes the threat, then player i 1 s best response is to pay the $1,000 But player i 1 should not believe the threat, because it is not credible: If player i2 were given the opportunity to carry out the threat, player i 2 would choose not to carry it out Player i 1 should pay player i 2 nothing V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 4 / 47

The framework We analyze in this chapter the following class of dynamic games with complete and perfect information There are 2 players and 2 moves First, player i 1 moves Then player i 2 observes player i 1 s move Then player i 2 moves and the game ends V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 5 / 47

Description of a specific class of games 1 Player i 1 chooses an action a i1 from a feasible set A i1 2 Player i 2 observes a i1 and then chooses an action a i2 from a feasible set A i2 3 Payoffs are u i1 (a i1, a i2 ) and u i2 (a i1, a i2 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 6 / 47

Description of a specific class of games Many economic problems fit this description Player i 2 s feasible set of actions A i2 could be allowed to depend on player i 1 s action a i1 such dependence could be denoted by A i2 (a i1 ) or could be incorporated into player i2 s payoff function, by setting u i2 (a i1, a i2 ) = for values of a i2 that are not feasible for a given a i1 Some moves by player i 1 could even end the game without player i 2 getting a move for such values of ai1, the set of feasible actions A i2 (a i1 ) contains only one element Other economic problems can be modeled by allowing for a longer sequence of actions either by allowing more players or by allowing players to move more than once V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 7 / 47

Other dynamic games with complete and perfect information The key features of a dynamic game of complete and perfect information are that 1 the moves occur in sequence 2 all previous moves are observed before the next move is chosen 3 the players payoffs from each feasible combination of moves are common knowledge V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 8 / 47

Backwards induction We solve a game from this class by backwards induction as follows: When player i 2 gets the move at the second stage of the game He will face the following problem Given the action a i1 previously chosen by player i 1 : argmax{u i2 (a i1, a i2 ) : a i2 A i2 } Assume that for each a i1 in A i1, player i 2 s optimization problem has a unique solution, denoted by R i2 (a i1 ) This is player i 2 s reaction (or best response) to player i 1 s action V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 9 / 47

Backwards induction Recall that payoffs are common knowledge Therefore player i 1 can solve i 2 s problem as well as i 2 can Player i 1 will anticipate player i 2 s reaction to each action a i1 i 1 might take Thus player i 1 s problem at the first stage amounts to that argmax{u i1 (a i1, R i2 (a i1 )) : a i1 A i1 } Assume that the previous optimization problem for i 1 also has a unique solution, denoted by a i 1 The pair of actions (a i 1, R i2 (a i 1 )) is called the backwards induction outcome of this game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 10 / 47

Backwards induction and credible threats The backwards induction outcome does not involve non-credible threats Player i 1 anticipates that player i 2 will respond optimally to any action a i1 that i 1 might choose, by playing R i2 (a i1 ) Player i 1 gives no credence to threats by player i 2 to respond in ways that will not be in i 2 s self-interest when the second stage arrives V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 11 / 47

A 3-move game Consider the following 3-move game in which player i 1 moves twice 1 Player i 1 chooses L or R L ends the game with payoffs of 2 to i 1 and 0 to i 2 2 Player i 2 observes i 1 s choice: if i 1 chose R then i 2 can choose between L and R L ends the game with payoffs of 1 to both players 3 Player i 1 observes i 2 s choice 1 :if the earlier choices were R and R then i 1 chooses L of R, both of which end the game L with payoffs of 3 to player i 1 and 0 to player i 2 R with payoffs of 0 to player i 1 and 2 to player i 2 1 And recalls his own choice in the first stage V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 12 / 47

A 3-move game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 13 / 47

A 3-move game Let s compute the backwards induction outcome of this game We begin at the third stage, i.e., player i 1 s second move The strategy L is optimal At the second stage, player i 2 anticipates that if the game reaches the third stage then i 1 will play L Payoff of 1 from action L Payoff of 0 from action R At the second stage, the optimal action for player i 2 is L At the first stage, player i 1 anticipates that if the game reaches the second stage then i 2 will play L Payoff of 2 from action L Payoff of 1 from action R The first stage choice for player i 1 is L, thereby ending the game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 14 / 47

Consistency problem In the second stage i 2 anticipates that if the game reaches the third stage then i 1 will play L By doing this, i 2 is assuming that i 1 is rational This seems inconsistent with the fact that i 2 gets to move in the second stage only if i 1 deviates from the backwards induction outcome It may seem that if player i 1 plays R in the first stage then i 2 cannot assume in the second stage that i 1 is rational If i 1 plays R in the first stage then it cannot be common knowledge that both players are rational It is CK that the players are rational if all the players are rational, and all the players know that all the players are rational, and all the players know that all the players know that all the players are rational, and so on, ad infinitum V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 15 / 47

Solutions to the consistency problem There remain reasons for i 1 to have chosen R that do not contradict i 2 s assumption that i 1 is rational Solution 1 It is CK that i 1 is rational but not that player i 2 is rational Both players are rational but if i 1 thinks that i 2 might not be rational Then i 1 might choose R in the first stage Hoping that i 2 will play R in the second stage Thereby giving i 1 a chance to play L in the second stage V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 16 / 47

Solutions to the consistency problem Solution 2 It is CK that i 2 is rational but not that i 1 is rational Both players are rational but if i 1 thinks that i 2 thinks that i 1 might not be rational Then i 1 might choose R in the first stage Hoping that i 2 will think that i 1 is not rational And so play R in the hope that i 1 will play R in the third stage Backwards induction assume that i 1 s choice of R could be explained along these lines V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 17 / 47

Stackelberg model of duopoly Stackelberg (1934) proposed a dynamic model of duopoly A dominant (leader) firm moves first A subordinate (follower) firm moves second At some points in the history of the U.S. automobile history, for example, General Motors has seemed to play such a leadership role As in the Cournot model, Stackelberg assumes that firms choose quantities V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 18 / 47

Timing of the game 1 Firm i 1 chooses the quantity q i1 0 2 Firm i 2 observes q i1 and then chooses a quantity q i2 0 3 The payoff to firm i is given by the profit function π i (q i, q j ) = q i [P (Q) c] where P (Q) = [a Q] + is the market-clearing price when the aggregate quantity on the market is Q = q i1 + q i2 c is the constant marginal cost of production (no fixed costs) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 19 / 47

Solving by backwards induction We first compute i 2 s reaction to an arbitrary quantity of i 1 R i2 (q i1 ) argmax{π i2 (q i1, q i2 ) : q i2 0} which yields R i2 (q i1 ) = a c q i1 2 if q i1 < a c 0 if q i1 a c Second, i 1 can solve i 2 s problem as well as i 2 can solve it Firm i 1 should anticipate that the quantity choice q i1 will be met with the reaction R i2 (q i1 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 20 / 47

Solving by backwards induction Firm i 1 s problem in the first stage of the game amounts to argmax{π i1 (q i1, R i2 (q i1 )) : q i1 0} The backwards induction outcome of the Stackelberg duopoly game is (q i 1, q i 2 ) where q i 1 = a c 2 and q i 2 = R i2 (q i 1 ) = a c 4 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 21 / 47

Interpretation In the Nash equilibrium of the Cournot game (simultaneous moves) each firm produces (a c)/3 Thus aggregate quantity in the backwards induction outcome of the Stackelberg game, 3(a c)/4, is greater than in the Nash equilibrium So the market clearing price is lower in the Stackelberg game In the Stackelberg game, i 1 could have chosen its Cournot quantity, (a c)/3 In which case i 2 would have responded with its Cournot quantity V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 22 / 47

Interpretation In the Stackelberg game, i 1 could have achieved its Cournot profit level but chose to do otherwise So i 1 s profit in the Stackelberg game must exceed its profit in the Cournot game The market clearing price is lower in the Stackelberg game We can prove that the aggregate profits are lower with respect to the Cournot outcome Therefore, the fact that i 1 is better off implies that i 2 is worse off V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 23 / 47

Interpretation In game theory, having more information can make a player worse off More precisely, having it known to the other players that one has more information can make a player worse off In the Stackelberg game, the information in question is i 1 s quantity Firm i 2 knows i 1 s action q i1 And firm i 1 knows that i 2 knows q i1 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 24 / 47

Modifying the sequential move game Assume know that firm i 1 chooses first q i1, after which i 2 chooses q i2 but does so without observing q i1 If i 2 believes that i 1 has chosen its Stackelberg quantity q i 1 = (a c)/2 Then i 2 s best response is again R i2 (q i 1 ) = (a c)/4 But if i 1 anticipates that i 2 will hold this belief and so choose q i 2 Then i 1 prefers to choose its best response to (a c)/4, namely, 3(a c)/8 rather than its Stackelberg quantity (a c)/2 Thus firm i 2 should not believe that i 1 has chosen its Stackelberg quantity Actually this game coincides with the simultaneous Cournot game The unique Nash equilibrium is for both firms to choose the quantity (a c)/3 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 25 / 47

Wages and employment in a unionized firm Leontief (1946) proposed the following model of the relationship between a firm and a monopoly union The union is the monopoly seller of labor to the firm The union has exclusive control over wages But the firm has exclusive control over employment The union s utility function is U(w, L) where w 0 is the wage the union demands from the firm L 0 is employment We assume that (w, L) U(w, L) is increasing in both w and L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 26 / 47

Wages and employment in a unionized firm The firm s profit function is π(w, L) R(L) wl where R(L) is the revenue the firm can earn if it employs L workers We assume that L R(L) is twice continuously differentiable, strictly increasing (i.e., R > 0), strictly concave (i.e., R < 0) and satisfies Inada s condition at 0 and, i.e., lim L 0 R (L) = + and lim + L R (L) = 0 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 27 / 47

Timing of the game 1 The union makes a wage demand, w 2 The firm observes and accepts w and then chooses employment, L 3 Payoffs are U(w, L) and π(w, L) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 28 / 47

Backwards induction outcome of the game First, we can characterize the firm s best response L (w) in stage 2 to an arbitrary wage demand w by the union in stage 1 Given w the firm chooses L (w) to solve L (w) argmax{π(w, L) : L 0} If w > 0 then there is a unique solution L (w) satisfying R (L (w)) = w V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 29 / 47

Backwards induction outcome of the game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 30 / 47

Firm s isoprofit curves Fixing the wage level w on the vertical line, the firm s choice of L amounts to the choice of a point on the horizontal line {(L, w) : L 0} The highest feasible profit level is attained by choosing L such that the isoprofit curve through (L, w) is tangent to the constraint {(L, w) : L 0} Holding L fixed, the firm does better when w is lower V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 31 / 47

Union s indifference curves Holding L fixed, the union does better when w is higher Higher indifference curves represent higher utility levels for the union V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 32 / 47

Backwards induction outcome We turn to the union s problem at stage 1 The union can solve the firm s second stage problem as well as the firm can solve it The union should anticipate that the firm s reaction to the wage demand w will be to choose the employment level L (w) Thus, the union s problem at stage 1 amounts to solve argmax{u(w, L (w)) : w > 0} The union would like to choose the wage demand w that yields the outcome (w, L (w)) that is on the highest possible indifference curve V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 33 / 47

Backwards induction outcome The solution to the union s problem, w, is the wage demand such that the union s indifference curve through the point (L (w ), w ) is tangent to the curve {L (w) : w > 0} at that point V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 34 / 47

Inefficiency The backwards induction outcome (w, L (w )) is inefficient Both the union s utility and the firm s profit would be increased it (L, w) were in the shaded region V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 35 / 47

Repeated games Espinosa and Rhee (1989) propose one answer to this puzzle Based on the fact that the union and the firm negotiate repeatedly over time There may exist an equilibrium of such a repeated game in which the union s choice of w and the firm s choice of L lie in the shaped region V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 36 / 47

Sequential bargaining Two players are bargaining over one dollar They alternate in making offers First player i 1 makes a proposal that i 2 can accept or reject If i 2 rejects then i 2 makes a proposal that i 1 can accept and reject And so on Each offer takes one period, and the players are impatient They discount payoffs received in later periods by a factor δ (0, 1) per period V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 37 / 47

Discount factor The discount factor δ reflects the time-value of money A dollar received at the beginning of one period can be put in the bank to earn interest, say at rate r per period So this dollar will be worth 1 + r dollars at the beginning of the next period Equivalently, a dollar to be received at the beginning of the next period is worth only 1/(1 + r) of a dollar now Let δ = 1/(1 + r) Then a payoff π to be received in the next period is worth only δπ now A payoff π to be received two periods from now is worth only δ 2 w now, and so on The value today of a future payoff is called the present value of that payoff V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 38 / 47

Timing of 3-period bargaining game (1a) At the beginning of the first period, player i 1 proposes to take a share s 1 of the dollar, leaving 1 s 1 for player i 2 (1b) Player i 2 either accepts the offer, in which case the game ends and the payoffs s 1 to i 1 and 1 s 1 to i 2 are immediately received rejects the offer, in which case play continues to the second period (2a) At the beginning of the second period, i 2 proposes that player i 1 take a share s 2 of the dollar, leaving 1 s 2 for i 2 Observe that s t always goes to player i 1 regardless of who made the offer (2b) Player i 1 either accepts the offer, in which case the game ends and the payoffs s2 to i 1 and 1 s 2 to i 2 are immediately received rejects the offer, in which case play continues to the third period (3) At the beginning of the third period, i 1 receives a share s of the dollar, leaving 1 s for player i 2, where s (0, 1) is exogenously given V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 39 / 47

Backwards induction outcome We first compute i 2 s optimal offer if the second period is reached Player i 1 can receive s in the third period by rejecting i 2 s offer of s 2 this period But the value this period of receiving s next period is only δs Thus, i 1 will accept s2 if s 2 δs reject s2 if s 2 < δs We assume that each player will accept an offer if indifferent between accepting and rejecting V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 40 / 47

Backwards induction outcome Player i 2 s decision problem in the second period amounts to choosing between receiving 1 δs this period by offering s2 = δs to player i 1 receiving 1 s next period by offering player i1 any s 2 < δs The discounted value of the latter decision is δ(1 s), which is less than 1 δs available from the former option So player i 2 s optimal second-period offer is s 2 = δs Thus, if play reaches the second period, player i 2 will offer s 2 and player i 1 will accept V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 41 / 47

Backwards induction outcome Since i 1 can solve i 2 s second-period problem as well as player i 2 can Then i 1 knows that i 2 can receive 1 s 2 in the second period by rejecting i 1 s offer of s 1 this period The value this period of receiving 1 s 2 next period is only δ(1 s 2 ) Thus player i 2 will accept i 1 s offer of s 1 this period if and only if 1 s 1 δ(1 s 2) or s 1 1 δ(1 s 2) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 42 / 47

Backwards induction outcome Player i 1 s first-period decision problem therefore amounts to choosing between receiving 1 δ(1 s 2 ) this period by offering 1 s 1 = δ(1 s 2) to i 2 receiving s 2 next period by offering 1 s 1 < δ(1 s 2) to i 2 The discounted value of the latter option is δs 2 = δ2 s which is less than the 1 δ(1 s 2 ) = 1 δ(1 δs) available from the former option Thus player i 1 s optimal first-period offer is s 1 = 1 δ(1 s 2 ) = 1 δ(1 δs) The backwards induction outcome of this 3-period game is i 1 offers the settlement (s 1, 1 s 1) to i 2, who accepts V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 43 / 47

The infinite horizon case The timing is as described previously Except that the exogenous settlement in step (3) is replaced by an infinite sequence of steps (3a), (3b), (4a), (4b), and so on Player i 1 makes the offer in odd-numbered periods, player i 2 in even-numbered Bargaining continues until one player accepts an offer We would like to solve backwards Because the game could go on infinitely, there is no last move at which to begin such an analysis V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 44 / 47

The infinite horizon case A solution was proposed by Shaked and Sutton (1984) The game beginning in the third period (should it be reached) is identical to the game as a whole (beginning in the first period) In both cases (game beginning in the third period or as a whole) player i 1 makes the first offer the players alternate in making subsequent offers the bargaining continues until one player accepts an offer V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 45 / 47

The infinite horizon case Suppose that there is a backwards induction outcome of the game as a whole in which players i 1 and i 2 receive the payoffs s and 1 s We can use these payoffs in the game beginning in the third period, should it be reached And then work backwards to the first period, as in the 3-period model, to compute a new backwards induction outcome for the game as a whole In this new backwards induction outcome, i 1 will offer the settlement (f(s), 1 f(s)) in the first period and i 2 will accept, where f(s) = 1 δ(1 δs) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 46 / 47

The infinite horizon case Let s H be the highest payoff player i 1 can achieve in any backwards induction outcome of the game as a whole Using s H as the third-period payoff to player i 1, this will produce a new backwards induction outcome in which player i 1 s first-period payoff is f(s H ) Since s f(s) = 1 δ + δ 2 s is increasing, the payoff f(s H ) must coincide with s H The only value of s that satisfies f(s) = s is 1/(1 + δ), which will be denoted by s Actually we can prove that (s, 1 s ) is the unique backwards-induction outcome of the game as a whole In the first period, i 1 offers the settlement (s, 1 s ) Player i 2 accepts V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 47 / 47