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Game Theory and Strategy

Game Theory and Strategy. - Week 10 - Instructor: Dr Shino Takayama. Agenda for Week 10. Chapter 6 Stackelberg’s model Buying votes Chapter 7 Allowing for simultaneous moves Entry into a monopolized industry. Stackelberg’s Model of Duopoly.

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Game Theory and Strategy

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  1. Game Theory and Strategy - Week 10 - Instructor: Dr Shino Takayama

  2. Agenda for Week 10 • Chapter 6 • Stackelberg’s model • Buying votes • Chapter 7 • Allowing for simultaneous moves • Entry into a monopolized industry

  3. Stackelberg’s Model of Duopoly • Consider a market in which there are two firms, both producingthe same good. Firm i's cost of producing qi units of the good is Ci(qi); the price atwhich output is sold when the total output is Q is Pd(Q). • Each firm's strategic variable is output, as in Cournot'smodel, but the firms make their decisions sequentially, rather thansimultaneously: one firm chooses its output, then the other firm does so, knowingthe output chosen by the first firm.

  4. Example: Constant unit cost and linear inverse demand • Each firm’s cost function is given by Ci(qi) = cqi. • The inverse demand function is given by: α− Q if Q ≤ α P(Q) = 0 if Q > α, where α > 0 and c > 0 are constant.

  5. Backward induction: Firm 2 • The firms’ payoffs are: π2(q1, q2) = q2(P(q1+q2) – c) q2(α− c − q1 − q2) if q1+q2≤ α; = − cq2 if q1+q2 > α. • We obtain the best response as: ½(α− c− q1) if q1≤ α – c; b2(q1) = 0if q1 > α - c.

  6. Backward induction: Firm 1 • Firm 1’s strategy is the output q1 that maximizes π1(q1, q2) = q1(α- c - (q1 + 1/2 (α- c - q1))) = 1/2 q1(α- c - q1). • Its maximizer is q1 = ½(α- c).

  7. Contrast with Cournot’s model • The unique subgame perfect equilibrium is: q1* = ½(α− c)and q2* = ¼(α− c). • π1(q1*, q2*) = q1*(Pd(q1*+ q2*) - c) = 1/8(α− c)2 • π2(q1*, q2*) = q2(Pd(q1*+q2*) - c) = 1/16(α− c)2 • Cournot’s case: q1C = q2C= 1/3(α− c) π1(q1C, q2C) = π2(q1C, q2C) = 1/9(α− c)2

  8. Buying Votes • A legislature has k members, where k is an odd number. • Two rival bills, X andY, are being considered. The bill that attracts the votes of a majority of legislatorswill pass. Interest group X favors bill X, whereas interest group Y favors bill Y. • Each group wishes to entice a majority of legislators to vote for its favorite bill. • Both interest groups give an amount of money (possibly zero) to each legislator. • Each interest group wishes to spend as little aspossible. • Group X values the passing of bill X at $VX > 0 and the passing of bill Yat zero, and group Y values the passing of bill Y at $VY > 0 and the passing ofbill X at zero.

  9. Buying votes: Set-up • Players: The two interest groups, X and Y. • Terminal histories: The set of all sequences (x, y), where x is a list of payments tolegislators made by interest group X and y is a list of payments to legislatorsmade by interest group Y. • Player function: P(φ) = X and P(x) = Y for all x.

  10. Buying votes: Set-up, continued • Preferences: The preferences of interest group X are represented by the payofffunction VX– (x1 + … + xk)if bill X passes – (x1 + … + xk)if bill Y passes, where bill Y passes after (x, y) if and only if the numberof components of y that are at least equal to the corresponding componentsof x is at least 1/2 (k + 1) . • The preferencesof interest group Y are represented by the analogous function (whereVY replaces VX, y replaces x, and Y replaces X).

  11. Group Y's best response • Let μ = 1/2 (k + 1), a bare majority of k legislators, and denote by mx the sum of thesmallest μcomponents of x -- the total payments Y needs to make to buy off a baremajority of legislators. • If mx < VY, then group Y can buy off a bare majority of legislators for lessthan VY, so that its best response to x is to match group X's payments to theμlegislators to whom group X's payments are smallest; the outcome is thatbill Y is passed. • If mx> VY, then the cost to group Y of buying off any majority of legislatorsexceeds VY, so that group Y's best response to x is to make no payments; theoutcome is that bill X is passed. • If mx= VY, then both the actions in the previous two cases are best responsesby group Y to x.

  12. What should group X do? • If it chooses a list of payments x for which mx < VY, then group Y matches its payments to a bare majority of legislators, and bill Ypasses. • If it reduces all its payments, the same bill is passed. • Thus the only list ofpayments x with mx< VYthat may be optimal is x = (0, . . . , 0). • If it chooses a list ofpayments x with mx> VY, then group Y makes no payments, and bill X passes. • If it reduces all its payments a little (keeping the payments to every bare majoritygreater than VY), the outcome is the same. • Thus no list of payments x for whichmx> VY is optimal.

  13. Property in Subgame Perfect Equilibrium • We conclude that in any subgame perfect equilibrium we have either 1. x =(0, . . . , 0) or 2. mx= VY. • How much does X need to pay to deter Y? • It needs to spend more than VY to every bare majority of legislators (more than VY/μfor each legislator). Thus, more than kVY/μin total. • If VX < kVY/μ, group X is better off making no payments than gettingbill X passed by making payments large enough to deter group Y from matchingits payments to a bare majority of legislators. • If VX> kVY/μ, group X can afford to make payments largeenough to deter group Y from matching.

  14. Suppose that VX ≠ kVY/μ. • The game has a unique subgame perfectequilibrium, in which group Y's strategy is: • match group X's payments to the mxlegislators to whom X's payments aresmallest after a history x for which mx < VY • make no payments after a history x for which mx> VY and group X's strategy depends on the relative sizes of VX and VY: • if VX< kVY/μ,then group X makes no payments; • if VX> kVY/μ,then group X pays each legislator VY/μ.

  15. Chapter 7 • A moregeneral model that allows us to study situations in which, after some sequences ofevents, the members of a group of decision-makers choose their actions simultaneously, each member knowing every decision-maker's previous actions, but notthe contemporaneous actions of the other members of the group.

  16. An extensive game with perfect information and simultaneousmoves • a set of players • a set of sequences (terminal histories) with the property that no sequence isa proper subhistory of any other sequence • a function (the player function) that assigns a set of players to every sequencethat is a proper subhistory of some terminal history • for each proper subhistory h of each terminal history and each player i thatis a member of the set of players assigned to h by the player function, a setAi(h) (the set of actions available to player i after the history h) • for each player, preferences over the set of terminal histories

  17. Example: BoS • First, person 1 decides whether to stay home andread a book or to attend a concert. • If she reads a book, the game ends. • If shedecides to attend a concert then, as in BoS, she and person 2 independently choosewhether to sample the aural delights of Bach or Stravinsky, not knowing the otherperson's choice.

  18. Set-up: Bos • Players:The two people (1 and 2). • Terminal histories:Book, (Concert, (B, B)), (Concert, (B, S)), (Concert, (S, B)), (Concert, (S, S)). • Player function: P(φ) = 1 and P(Concert) = {1, 2}. • Actions: A1(φ) = {Concert,Book} and A1 (Concert) = {B, S};A2(Concert) = {B, S}. • Preferences: u1-- (Concert, (B, B)) > Book > (Concert, (S, S)) > (Concert, (B, S)) ~ (Concert, (S, B)). u2-- (Concert, (S, S)) > Book > (Concert, (B, B)) > (Concert, (B, S)) ~ (Concert, (S, B)).

  19. Subgame Perfect Equilibrium • The game has three pure Nash equilibria: ((Concert, B), B), ((Book, B), S), and ((Book, S), S). • In the subgame that follows the history Concert, there are two Nash equilibria(in pure strategies), namely (S, S) and (B, B). • If the outcome in the subgame that follows Concert is (S, S) then the optimalchoice of player 1 at the start of the game is Book. • If the outcome in the subgame that follows Concert is (B, B) then the optimalchoice of player 1 at the start of the game is Concert. • The game has two subgame perfect equilibria: ((Book, S), S) and((Concert, B), B).

  20. Entry into a monopolized industry • An industry is currently monopolized by a single firm (the incumbent). A second firm (the challenger) is considering entry, which entails a positive cost fin addition to its production cost. If the challenger stays out then its profit iszero, whereas if it enters, the firms simultaneously choose outputs (as in Cournot'smodel of duopoly). The cost to firm i of producing qi units of outputis Ci(qi). If the firms' total output is Q then the market price is Pd(Q).

  21. Set-up • Players:The incumbent (firm 1) and the challenger (firm 2). • Terminal histories: (In, (q1, q2)) for any pair (q1, q2)of outputs (nonnegative numbers),and (Out, q1) for any output q1. • Player function: P(φ) = {2}, P(In) = {1, 2}, and P(Out) = {1}. • Actions: A2(φ) = {In, Out}; A1(In), A1(Out), and A2(In) are all equal to theset of possible outputs (nonnegative numbers). • Preferences:Each firm's preferences are represented by its profit, which for aterminal history (In, (q1, q2)) is q1Pd(q1 +q2) - C1(q1) for the incumbent andq2Pd(q1 +q2) - C1(q1) – ffor the challenger, and for a terminal history (Out, q1)is q1Pd(q1 +q2) - C1(q1) for the incumbent and 0 for the challenger.

  22. Extensive Form Game

  23. Example: Constant unit cost and linear inverse demand • Each firm’s cost function is given by Ci(qi) = cqi. • The inverse demand function is given by: α− Q if Q ≤ α P(Q) = 0 if Q > α, where α > 0 and c > 0 are constant.

  24. Nash equilibrium after In • Consider the subgame that follows the history In. • The strategic form of this subgame is thesame as the example of Cournot's duopoly game, exceptthat the payoff of the challenger is reduced by f regardlessof the challenger's output. • Thus the subgame has a unique Nash equilibrium, inwhich the output of each firm is 1/3 (α- c); • the incumbent's profit is 1/9(α− c)2 , and • the challenger's profit is 1/9(α− c)2- f .

  25. Nash equilibrium after Out • The incumbent’s profit is: q1(α− q1)− cq1 = q1 (α− c −q1). • The optimal action should be: q1 = 1/2 (α− c).

  26. Challenger’s action & Subgame Perfect Equilibrium • If f < 1/9(α− c)2, there is a unique subgame perfect equilibrium, in which thechallenger enters. The outcome is that the challenger enters and each firmproduces 1/3(α− c). • If f > 1/9(α− c)2, there is a unique subgame perfect equilibrium, in which thechallenger stays outand theincumbent produces 1/2(α− c). • If f = 1/9(α− c)2, the game has two subgame perfect equilibria: the one for thecase f < 1/9(α− c)2 and the one for the case f > 1/9(α− c)2.

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