Introduction to Game Theory

Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 2. Two-stage games of complete but imperfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 1 / 42

Topics covered 1 Theory: Subgame perfection 2 Bank runs 3 Tariffs and imperfect international competition 4 Tournaments V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 2 / 42

Theory: Subgame perfection We continue to assume that play proceeds in a sequence of stages The moves in all previous stages are observed before the next stage begins However, we now allow there be simultaneous moves within each stage The game has imperfect information V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 3 / 42

Theory: Subgame perfection We will analyze the following simple game: 1 Players i 1 and i 2 simultaneously choose actions a i1 and a i2 from feasible sets A i1 and A i2, respectively 2 Players i 3 and i 4 observe the outcome of the first stage, (a i1, a i2 ), and then simultaneously choose actions a i3 and a i4 from feasible sets A i3 and A i4, respectively 3 Payoffs are u i (a i1,..., a i4 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 4 / 42

Theory: Subgame perfection The feasible action sets of players i 3 and i 4 in the second stage, A i3 and A i4, could be allowed to depend on the outcome of the first stage, (a i1, a i2 ) In particular, there may be values of (a i1, a i2 ) that end the game One could allow for a longer sequence of stages either by allowing players to move in more than one stage or by adding players In some applications, players i 3 and i 4 are players i 1 and i 2 In other applications, either player i 2 or player i 4 is missing V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 5 / 42

Theory: Subgame perfection We solve the game by using an approach in the spirit of backwards induction The first step in working backwards from the end of the game involves solving a simultaneous-move game between players i 3 and i 4 in stage 2, given the outcome of stage 1 We will assume that for each feasible outcome (a i1, a i2 ) of the first game, the second-stage game that remains between players i 3 and i 4 has a unique Nash equilibrium denoted by (â i3 (a i1, a i2 ), â i4 (a i1, a i2 )) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 6 / 42

Theory: Subgame perfection If players i 1 and i 2 anticipate that the second-stage behavior of players i 3 and i 4 will be given by the functions â i3 and â i4 Then the first-stage interaction between i 1 and i 2 amounts to the following simultaneous-move game 1 Players i 1 and i 2 simultaneously choose actions a i1 and a i2 from feasible sets A i1 and A i2, respectively 2 Payoffs are u i (a i1, a i2, â i3 (a i1, a i2 ), â i4 (a i1, a i2 )) Suppose (a i 1, a i 2 ) is the unique Nash equilibrium of this simultaneous-move game We will call (a i 1, a i 2, a i 3, a i 4 ) the subgame-perfect outcome of this two-stage game, where a i 3 = â i3 (a i 1, a i 2 ) and a i 4 = â i4 (a i 1, a i 2 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 7 / 42

Theory: Subgame perfection Attractive feature Players i 1 and i 2 should not believe a threat by players i 3 and i 4 that the latter will respond with actions that are not a Nash equilibrium in the remaining second-stage game Because when play actually reaches the second stage at least one of i 3 and i 4 will not want to carry out such a threat exactly because it is not a best response Unattractive feature Suppose player i 1 is also player i 3 and that player i 1 does not play a i 1 in the first stage Player i 4 may then want to reconsider the assumption that player i 3 (i.e., player i 1 ) will play â i3 (a i1, a i2 ) in the second stage V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 8 / 42

Bank runs: Diamond and Dybvig, J. of Political Economy, 1983 Two investors have each deposited D with a bank The bank has invested the deposits 2D in a long-term project If the bank is forced to liquidate its investment before the project matures, a total of α(2d) can be recovered, where 1 2 < α < 1 If the bank allows the investment to reach maturity, the project will pay out a total of β(2d), where β > 1 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 9 / 42

Bank runs There are two dates at which investors can make withdrawals from the bank date 1 is before the bank s investment matures date 2 is after For simplicity we assume that there is no discounting If both investors make withdrawals at date 1 then each receives αd and the game ends If only one investor makes a withdrawal at date 1 then that investor receives the whole deposit D, the other receives (2α 1)D, and the game ends V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 10 / 42

Bank runs Finally, if neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2 If both investors make withdrawals at date 2 then each receives βd > D and the game ends If only one investor makes a withdrawal at date 2 then that investor receives (2β 1)D > βd, the other receives D, and the game ends Finally if neither investor makes a withdrawal at date 2 then the bank returns βd to each investor and the game ends V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 11 / 42

Bank runs withdraw don t withdraw αd, αd D, (2α 1)D don t (2α 1)D, D next stage Date 1 withdraw don t withdraw βd, βd (2β 1)D, D don t D, (2β 1)D βd, βd Date 2 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 12 / 42

Bank runs To analyze this game, we work backwards Consider the normal-form game at date 2 The strategy withdraw strictly dominates don t withdraw There is a unique Nash equilibrium in this game: both investors withdraw, leading to a payoff of (βd, βd) Since there is no discounting, we can simply substitute this payoff into the normal-form game at date 1 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 13 / 42

Bank runs withdraw don t withdraw αd, αd D, (2α 1)D don t (2α 1)D, D βd, βd Date 1 This one period version of the two-period game has two pure strategy Nash equilibria: 1 both investors withdraw, leading to a payoff of (αd, αd) 2 both investors do not withdraw, leading to a payoff of (βd, βd) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 14 / 42

Bank runs The original 2-period bank runs game has two subgame perfect outcomes 1 both investors withdraw at date 1, yielding payoffs of (αd, αd) 2 both investors do not withdraw at date 1 but do withdraw at date 2, yielding payoffs of (βd, βd) at date 2 The first of these outcomes can be interpreted as a run on the bank If investor i1 believes that investor i 2 will withdraw at date 1 then investor i 1 s best response is to withdraw, even though both investors would be better off if they waited until date 2 to withdraw Since there are two subgame perfect equilibria, this model does not predict when bank runs will occur, but does show that they can occur as an equilibrium phenomenon V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 15 / 42

Tariffs and imperfect international competition Consider two identical countries, denoted by i 1 and i 2 Each country has a government that chooses a tariff rate a firm that produces output for both home consumption and export consumers who buy on the home market from either the home firm or the foreign firm 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 ] + V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 16 / 42

Tariffs and imperfect international competition The firm in country i (called firm i) produces h i for home consumption and e i for export, in particular we have Q i = h i + e j The firms have a constant marginal cost c and no fixed costs: we assume that c < a The total cost of production for firm i is C i (h i, e i ) c(h i + e i ) The firms also incur tariff costs on exports if firm i exports e i to country j when government j has set the tariff rate t j then firm i must pay tj e i to government j V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 17 / 42

Tariffs and imperfect international competition 1 The governments simultaneously choose tariff rates, t i1 and t i2 2 The firms observe the tariff rates and simultaneously choose quantities for home consumption and for export (h i, e i ) 3 Payoffs are profit to firms and total welfare to governments V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 18 / 42

Tariffs and imperfect international competition Profit to firm i is π i (t i, t j, h i, e i, h j, e j ) [a (h i + e j )] + h i + [a (e i + h j )] + e i c(h i + e i ) t j e i Total welfare to government i, where total welfare is the sum of consumers surplus enjoyed by the consumers in country i, the profit earned by the firm i, and the tariff revenue collected by government i from firm j W i (t i, t j, h i, e i, h j, e j ) 1 2 Q2 i + π i (t i, t j, h i, e i, h j, e j ) + t i e j V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 19 / 42

Tariffs and imperfect international competition Suppose the governments have chosen the tariffs t i1 and t i2 Assume that (h i 1, e i 1, h i 2, e i 2 ) is a Nash equilibrium in the remaining game between firms i 1 and i 2 Then, for each i, (h i, e i ) must solve argmax{π i (t i, t j, h i, e i, h j, e j) : h i 0 and e i 0} Firm i is maximizing profits on market i and market j h i must solve argmax{h i ( [a (hi + e j)] + c ) : h i 0} e i must solve argmax{e i ( [a (ei + h j)] + c t j ) : ei 0} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 20 / 42

Tariffs and imperfect international competition Assuming e j a c, we have h i = 1 2 (a e j c) Assuming h j a c t j, we have e i = 1 2 (a h j c t j ) We obtain four equations with four unknowns V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 21 / 42

Tariffs and imperfect international competition If t i (a c)/2 for each player i, then the solutions are h i (t i ) = a c + t i 3 and e i (t j ) = a c 2t j 3 In the Cournot game, both firms were choosing the quantity (a c)/3, but this result was derived under the assumption of symmetric marginal costs In the equilibrium described above, the governments tariff choices make marginal costs asymmetric On market i, firm i s marginal cost is c but firm j s is c + ti Since firm j s cost is higher it wants to produce less If firm j is going to produce less, then the market-clearing price will be higher, so firm i wants to produce more In equilibrium the function h i increases in t i and e j decreases (at a faster rate) in t i V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 22 / 42

Tariffs and imperfect international competition Having solved the second-stage game that remains between the two firms after the governments choose tariff rates We can now represent the first-stage interaction between the two governments as the following simultaneous-move game First, the governments simultaneously choose tariff rates t i1 and t i2 Second, payoffs are W i (t i, t j, h i (t i ), e i (t j ), h j(t j ), e j(t i )) We now solve for the Nash equilibrium of this game between the governments V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 23 / 42

Tariffs and imperfect international competition We denote by (t i, t j ) W i (t i, t j ) the function defined by W i (t i, t j ) W i (t i, t j, h i (t i ), e i (t j ), h j(t j ), e j(t i )) If (t i, t j ) is a Nash equilibrium of this game between governments then, for each i, the tariff t i must solve argmax{w i (t i, t j) : t i 0} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 24 / 42

Tariffs and imperfect international competition We propose to show that there exists a solution (t i, t j) (0, (a c)/2) (0, (a c)/2) Observe that if t i and t j equals belong to (0, (a c)/2) then W i (t i, t j ) (2(a c) t i ) 2 18 A solution is + (a c + t i) 2 9 + (a c 2t j )2 9 t i = a c 3 + t i(a c 2t i ) 3 for each i, independent of t j In this model, choosing a tariff rate of (a c)/3 is a dominant strategy for each government V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 25 / 42

Tariffs and imperfect international competition We then obtain the following firms quantity choices for the second-stage h i (t i ) = 4(a c) 9 and e i (t j) = a c 9 Thus, the subgame-perfect outcome of this tariff game is t i 1 = t i 2 = a c 3, h i 1 = h 4(a c) i 2 = 9 and e i 1 = e i 2 = a c 9 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 26 / 42

Tariffs and imperfect international competition If the governments had chosen tariff rates equal to 0 Then the aggregate quantity on each market would have been Q i = just as in the Cournot model 2(a c) 3 The consumers surplus on market i is lower when the governments choose their dominant strategy tariffs than it would be if they chose zero tariffs In fact, zero tariffs is socially optimal, i.e., it is the solution of argmax{w i 1 (t i1, t i2 ) + W i 2 (t i1, t i2 ) : t i1 0 and t i2 0} There is an incentive for the governments to sign a treaty in which they commit to zero tariffs V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 27 / 42

Tournaments: Lazear and Rosen, J. of Political Economy, 1981 Consider two workers J = {j 1, j 2 } and their boss Worker j produces output y j = e j + ε j, where e j is effort and ε j is noise Production proceeds as follows: 1 The workers simultaneously choose non-negative effort levels: e j 0 2 The noise terms ε j1 and ε j2 are independently drawn from a density f : R [0, ) with zero mean 3 The workers output are observed but their effort choices are not The workers wages therefore can depend on their outputs but not directly on their effort levels V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 28 / 42

Tournaments Suppose the boss decides to induce effort by having the workers compete in a tournament The winner of the tournament is the worker with the higher output The wage earned by the winner of the tournament is w H ; the wage earned by the loser is w L The payoff to a worker from earning wage w and expending effort e is u(w, e) = w g(e) where g(e) is the disutility corresponding to the effort level e The function g : [0, ) [0, ) is twice continuously differentiable and satisfies g > 0 (strictly increasing) and g > 0 (strictly convex) The payoff to the boss is y j1 + y j2 w H w L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 29 / 42

Tournaments The boss is player i 1 whose action a i1 paid in the tournament, w H and w L is choosing the wages to be There is no player i 2 Worker j 1 is player i 3 and worker j 2 is player i 4 Workers observe the wages chosen in the first stage and then simultaneously choose actions a i3 and a i4, namely effort choices e j1 and e j2 Since outpouts (and so also wages) are functions not only of the players actions but also of the noise term ε j1 and ε j2, we work with the players expected payoffs according to the density f V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 30 / 42

Tournaments Suppose that the boss has chosen the wages w H and w L Let (e j 1, e j 2 ) be a Nash equilibrium of the remaining game between the workers For each j, e j must solve argmax{π j (w H, w L, e j, e k ) : e j 0} where π j (w H, w L, e j, e k ) is the expected profit defined by π j (w H, w L, e j, e k ) = w H Prob{y j (e j ) > y k (e k )} +w L Prob{y j (e j ) < y k (e k )} g(e j) = (w H w L ) Prob{y j (e j ) > y k (e k )} +w L g(e j ) where y j (e j ) = e j + ε j and y k (e k ) = e k + ε k V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 31 / 42

Tournaments Assume e j is strictly positive The first-order condition of the maximization problem is (w H w L ) Prob{y j(e j ) > y k (e k )} e j = g (e j ) The worker j chooses e j such that the marginal disutility of extra effort, g (e j ), equals the marginal gain from extra effort V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 32 / 42

Tournaments Observe that by Bayes rule Prob{y j (e j ) > y k (e k )} = Prob{ε j > e k + ε k e j } = Prob{ε j > e k + z e j ε k = z}f(z)dz [0, ) Since ε j and ε k are independent we have implying that 1 Prob{ε j > e k + z e j ε k = z} = Prob{ε j > e k + z e j} Prob{y j (e j ) > y k (e k )} = R [1 F (e k e j + z)]f(z)dz The first order condition becomes (w H w L ) f(e k e j + z)f(z)dz = g (e j ) 1 F is the cumulative distribution of f R V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 33 / 42

Tournaments If we look for symmetric Nash equilibria we get e j = e k = e (w H, w L ) (w H w L ) f(z) 2 dz = g (e (w H, w L )) R Since g is convex, a bigger prize for winning (i.e., a larger value of w H w L ) induces more effort V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 34 / 42

Tournaments Holding the prize constant, it is not worthwhile to work hard when output is very noisy, because the outcome of the tournament is likely to be determined by luck rather than effort If ε j is normally distributed with variance σ 2, then f(z) 2 dz = 1 2σ π R which decreases in σ, so e (w H, w L ) decreases in σ V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 35 / 42

Tournaments We work backwards to the first stage of the game Suppose that if the workers agree to participate in the tournament (rather than accept alternative employment) Then they will respond to the wages w H and w L by playing the symmetric Nash equilibrium previously exhibited We ignore the possibility of asymmetric equilibria and of an equilibrium with corner solutions Suppose that the workers alternative employment opportunity would provide utility U a V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 36 / 42

Tournaments In the symmetric Nash equilibrium each worker wins the tournament with probability 1/2 Prob{y j (e (w H, w L )) > y k (e (w H, w L ))} = 1 2 If the boss intends to induce the workers to participate in the tournament then he must choose wages (w H, w L ) that satisfy 1 2 w H + 1 2 w L g(e (w H, w L )) U a (IR) The boss chooses wages to maximize expected profit 2e (w H, w L ) + E[ε 1 + ε 2 ] (w H + w L ) = 2e (w H, w L ) (w H + w L ) subject to the restriction (IR) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 37 / 42

Tournaments Assume there exists a solution (wh, w L ) to the maximization problem with wl > 0 The participation restriction (IR) must be binding at the optimum, i.e., (wh, w L ) must be a solution to w H + w L = 2U a + 2g(e (w H, w L )) (IRb) Expected profit becomes 2 [e (w H, w L) U a g(e (w H, w L))] The choice (wh, w L ) of the boss solves max w H w L 0 e (w H, w L ) g(e (w H, w L )) under the binding restriction (IRb) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 38 / 42

Tournaments We denote by f the function defined by δ 0, f (δ) = [g ] 1 (δξ) where ξ = Observe that e (w H, w L ) = f (w H w L ) R f(z) 2 dz We propose to replace the pair of variable (w H, w L ) by (δ, w L ) where δ = w H w L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 39 / 42

Tournaments It follows that the choice (δ, wl ) of the boss solves under the restriction max f (δ) g(f (δ)) (δ,w L ) 0 w L = U a + g(f (δ)) δ (IRt) Since the choice variable w L does not enter the objective function, the maximization problem is equivalent to the following one under the restriction max δ 0 f (δ) g(f (δ)) U a + g(f (δ)) δ 0 (IR ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 40 / 42

Tournaments Since we assumed that wl > 0 At the solution δ the restriction (IR ) is not binding U a + g(f (δ )) δ > 0 It follows that the choice δ of the boss satisfies the FOC Ψ (δ ) = 0 where the function Ψ is defined by Ψ(δ) f (δ) g(f (δ)) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 41 / 42

Tournaments Thus the optimal induced effort e (w H, w L ) satisfies Remember that (wh wl) g (e (w H, w L)) = 1 R f(z) 2 dz = g (e (w H, w L)) Therefore the optimal wages satisfy (wh wl) f(z) 2 dz = 1 The pair (wh, w L ) is determined by the participation equation wh + wl = 2U a + 2g ( [g ] 1 (1) ) R V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, 2011 42 / 42