2350gametheoryGames

2350gametheoryGames - GAME 1: THE “LOONIES” DILEMMA...

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Unformatted text preview: GAME 1: THE “LOONIES” DILEMMA Suppose you and a person you don’t know (never met and never will) are given the following instructions: Instructions; 0 Each of you has two options (or actions): SELF—INTERESTED: Request that $1,000 loonies be given to You Or ALTRUISTIC: Request that $2,000 loonies be given to the Other 0 Each person ’s payoff will be: THE SUM OF MONEY THAT WAS REQUESTED FOR THIS PERSON. 0 You must choose without knowing the other’s choice (same as, if the two of you choose simultaneously). Four Possible Outcomes: If you are SELF-INTERESTED and the other is ALTRUISTIC, you get 1000 + 2000 = 3000 and the other gets 0 + 0 = 0 If you are ALTRUISTIC and the other is SELF—INTERESTED, you get 0 + 0 = 0 and the other gets 1000 + 2000 = 3000 If you are both SELF-INTERESTED, you get: 1000 + 0 = 1000 and the other gets: 0 + 1000 = 1000 If you are both ALTRUISTIC, you get: 0 + 2000 = 2000 and the other gets: 2000 + 0 = 2000 The OUTCOME MATRIX for the above game specifies 0 The Players 0 Each player’s actions/strategies 0 The Payoffs (for each combination of strategies) Player II Each player here has a DOMINANT STRATEGY: S The EQUILIBRIUM of the GAME is: (s , S) with payoffs (1000, 1000) The outcome is inefficient; Both could be better off if each of them chose to be altruistic; then each would get more (2000>1000). LESS! m ; 0 The pursuit of self interest can lead to the ‘detriment’ of society. GAME 2: THE PRISONERS’ DILEMMA Two suspects in a big robbery are interrogated in separate cells. The prosecutor presents each of them with the following proposition: 0 If you CONFESS and the other DENIES the two of you robbed the bank, then you will be treated leniently and get a 6-month sentence. 0 If you Deny and the other Confesses then you will get 10 years. 0 If both of you confess, then each of you will get 8 years in prison. (A leniency of only 2 years since no single prisoner’s testimony is necessary to solve the crime given the other confessed. ) 0 If none of you confesses, then we will prosecute each of you for you past petty crimes already on your record and each will get 1 year in prison Prisoner II Prisoner I GAME 3: THE CARTEL PROBLEM Firms agree to coordinate their actions and restrict production (supply) in order to keep price high and thus make greater profit. Will each firm do as promised? 0 If both firms produce a large quantity (QL), then the price will fall substantially and each firm ’s profit will be low (10). 0 If each firm produces small quantity/(Q5 ), say, half the monopoly output, then they will share the monopoly profit ( 25 each). 0 If one firm supplies a large quantity and the other supplies a small quantity, then the first firm will make a larger profit at the expense of the other (because it will sell a lot at a relatively high price.) Firm II How can “cheating” be avoided? The Cournot Model of Quantity Competition Assumptions 0 There are two identical firms (No Entry): Firm 1 and Firm 2 0 They produce a homogeneous product. o Denote Firm 1’s output by ql and Firm 2’s output by qg. 0 Each firm has constant (average) cost: M C : AC : c. o Inverse Demand is given by p : A — BQ. 0 Each Firm realizes that if a total of Q : Q1 + Q2 units is produced then the market will clear at the price p : A — B (q1 + qg). 0 Each Firm chooses its own quantity without knowing the quantity of the other (that is, they choose quantities simultaneously). NOTE: Here the Strategies are the Quantities. This is a game in continuous strategies since each firm’s quantity can be any number from 0 to infinity. Solution: Finding the Nash Equilibrium (NE) The NE is a pair of quantities, (qf, (1;), such that the quantity produced by each firm is the best response to the quantity produced by the other firm. In symbols: BR1(q’2“) = q? and BR2<£1T> = q: Thus we have to find Best Response Function of each firm. For each firm, this function (or equation or formula) gives the best value for the firm’s quantity given the quantity of its rival. The BB functions can be represented graphically. The intersection of the BR curves is the Nash Equilibrium of the game. The solution of the two BR equations gives the equilibrium values of ql and q2. FINDIN THEBET EPNEFNTIN FFIMl: Inlax 7T1 : pqi — qu : [A — 3((11 + Q2)lqi — 0611 Set the derivative of 711 wrt ql equal to zero to get: 8 fl:A—2Bq qg—o:0 391 Solve for ql in terms of qg to get the Best Response Function of Firm 1: A — e 1 : — — 1 611 2B 2(12 ( ) NOTE: According to the above equation o If qg : 0 (Firm 2 produces nothing), it is best for Firm 1 to set ql : ‘4—C : 23 Monopoly Output (QM). O qg : _ 11111 pI‘O UCGS e eompe 1 1V8 011 p11 , 1 18 GS 01' If ABC F' 2 d th‘ t't' t t’ 't' b tf Firm 1 to set q1 : O. 0 Finally, the slope of the BR1 curve is —1/2. What does this mean? Thus the BRl curve is as shown below: I q2 QC FINDIN THEBET EPNEFNTIN FFIMZ: Iggx 7T2 : PQ2 — CQ2 : [A — 3((11 + Q2)lQ2 — 6612 Set the derivative of 7T2 wrt qg equal to zero to get: 8 fl:A_QBq2 . q1_c:0 3% Solve for qg in terms of ql to get the Best Response Function of Firm 2: A — e 1 : — — 2 Q2 2B 2(11 ( ) NOTE: According to the above equation o If ql : 0 (Firm 1 produces nothing), it is best for Firm 2 to set qg : é‘B‘: : Monopoly Output (QM). o If ql : Age (Firm 1 produces the ‘Competitive output’), it is best for Firm 2 to set q1 : O. 0 Finally, the slope of the BBQ curve is —1/2. What does this mean? Thus the BR2 curve is as shown below: 92 Qm THE NASH EQUILIBRIUM: INTERSECTION OF BR1 AND BR2 Nash (Cournot) Equilibrium q2* q1* THE NASH EQUILIBRIUM: ALGEBRAIC SOLUTION Solving equations (1) and (2) simultaneously gives * _1A—C ql : q2 — —3 B Therefore, the total output sold is 2 A—C >i<: >i< >i<i >i<:_ Q q1+q2 Q 3< B) and the equilibrium price is 1 2 Finally total industry prom and the consumers’ surplus is Profit* = (2/9)(A-c)"2 / B and C8* = (2/9)(A-c)"2 / B 4 MPA 1 N WITH PE FE T MPETITI N AND M l\' P LY: PC. Cournot Monopoly Price c %A + g6 Agrc Total Quantity A50 g (Age) % (Age) Total Profit 0 391230? £01230? Consumer Surplus %(A;)2 at??? éMgCV Total Surplus #50)? $1450? 304ch DeadWeight Loss 0 fimgy éMch Pm P* Qm Q* Monopoly QC MC Demand GAME 4: PRICE COMPETITION among OLIGOPOLISTS Two firms produce a similar product and compete in prices. 0 If both firms charge a low price (PL), then each firm ’s profit will be low (2 ). 0 If both firms charge the monopoly price (PM), then they will Share the monopoly profit (3 each). 0 If one firm charges the low price and the other the high (monopoly) price, then low-price firm will get most of the consumers and make a larger profit (4) at the expense of the other (who will make only I). Firm II GAME 5: COMPETITION with “LOWEST PRICE GUARANTEE” Competition is exactly as in GAME 4 except now each firm advertises: ? WE WILL NOT BE UNDERSOLD! F irm II What is the equilibrium price? Are consumers better off? GAME 6: BELL CURVE and THE STUDENTS’ DILEMMA Players: Two Students. Each can request 2, 5 or 10 marks to be added to his test grade. The student who requests the least will get 10 marks added and the other will get 0 marks added. But if they request the same amount, say X, then each will get X—2 -- PUBLIC GOODS Goods that are non-rival : (same unit can be consumed by many or all) PURE PUBLIC GOODS -r' n non-excludable: once the good is produced, no person can be excluded from enjoying (the “exclusion costs” are very high...) Examples: 0 Clean Air, 0 Clean Streets 0 Internet (Free Web Sites), Street Lights, TV and Radio Broadcast Signals, Police Protection, National Defense, Parents’ welfare to their children Public Goods need not be provided by governments. GAME 7: PUBLIC GOODS: The Neighbours Example 0 If each neighbour contributes $10 to charity then crime will be reduced Md each wilLsee as a result an morease in Mme equal to 12 (so each will receive a net payofi of 12—] 0:2). 0 If one contributes and the other doesn’t (free rides), then crime reduction is less and each will experience only an income increase of $6. 0 If no one contributes then nothing changes and so each gets 0. JerQ n ri Fr Ri Contribute Tom Free Ride GAME 8: PUBLIC GOODS: The Polluting Countries Example Country II RJduce Do Pollution Nothing Reduce Country I Pollution Do Nothing What can be done? ...
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This note was uploaded on 02/29/2012 for the course ECON 2350 taught by Professor Bardis during the Fall '12 term at York University.

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2350gametheoryGames - GAME 1: THE “LOONIES” DILEMMA...

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