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(1)ECO290E: Game Theory Lecture 4: Static Games with Continuous Strategies. 1. Lecture 4. 2013 Winter.

(2) Review of Lecture 3  . There are at least four reasons why we can expect Nash equilibrium (NE) would realize. By rational reasoning Being an outstanding choice A result of discussion A limit of some adjustment process. 1. 2. 3. 4. .  . In some cases, NE can be reached merely by rationality.    . 2. Focal point: Class room experiment Dominant strategies: Prisoner’s Dilemma Iterated elimination of strictly dominated strategies Lecture 4 2013 Winter.

(3) From Discrete to Continuous Strategies  . Discrete game: Players have a finite (called finite game) or countably many (inifinite) number of strategies.  .  . Checking inequalities reaches a Nash equilibrium.. Continuous game: Players have continuously many strategies, e.g., choosing a real number (price or quantity) from an interval.   . 3. Infinitely many inequalities; cannot be checked one by one… Use deduction (Bertrand model) or differential approach (Cournot model).. Lecture 4 2013 Winter.

(4) Bertrand Model Players: Two firms  Strategies: Prices they will charge  Payoffs: Profits  . Assumptions:  A linear demand function: P = a – bQ where b > 0  Common marginal cost, c. (c < a)  The firm with lower price must serve the entire market demand.  If the firms choose the same price, then each firm sells the half of the market demand. 4. Lecture 4 2013 Winter.

(5) Nash Equilibrium  . There is a unique NE in which both firms charge the price equal to their (common) marginal cost.. Why?  Choosing different prices never becomes a NE.  Choosing the same price other than the marginal cost also fails to be a NE.  If both firms choose p = c, then no firm has an (strict) incentive to change the price. ⇒ NE can also be solved by finding the intersection of best response/reply curves. 5. Lecture 4 2013 Winter.

(6) R[_ n. >” Best 2U .Oa"\n. Response Curves  =4U = =F U OQ¡§ U HA=! U. @U. m§ :$U 8U =5U. 67A. ,A. “”. ¾æ 6. ” “”. =TULuXæ. 4?:zf-Š' " ' &' 1ezKRL#g?zV Š. ]§. =!U. Lecture 4 2013 Winter.

(7) Bertrand Paradox Even if there are only two competitors, prices will be set at the level of marginal cost.  In reality, there are many industries that look like the Bertrand model but where prices are (much) higher than marginal cost.  There are at least three known explanations which can reasonably resolve this paradox:  .    . 7. Product differentiation Capacity constraints Dynamic interaction (collusion or cartel). Lecture 4 2013 Winter.

(8) Cournot Model Players: Two firms  Strategies: Quantities they will charge  Payoffs: Profits  . Assumptions:  A linear demand function: P = a - bQ  Common marginal cost, c.  Firms cannot decide their prices to charge, but the market price is determined so as to clear the market. ⇒ See Handout (“Oligopoly Competition” by Cabral). 8. Lecture 4 2013 Winter.

(9) Differential Approach  . Unlike Bertrand model, we CANNOT merely deduce an equilibrium. Some calculation is necessary, instead..  . Under Nash equilibrium, each player takes a best response against opponent’s strategy each other.    . 9. Given other player’s strategy, each player maximizes her payoff with respect to her own strategy. Can be solved by first order condition (which we call differential approach). Nash equilibrium is just a solution of simultaneous equations (of first order conditions).. Lecture 4 2013 Winter.

(10) b,aɑ7ɑɑ77 ɑDɑɑ1ɑDɑɑ*ɑ7g7Ɗɑaɑɑ*ɑg 9ɑ7 bgɑ Dɑɑɑ7ɑD*gɑ6ɑa7ɑg,aɑɑgb75ɑ¨ɑ7ɑ7 ɑ6ɑɑb,aɑ 1ɑDɑp*ɑ vc7ɑ ɑ D5ɑ r 7ɑ ɑ ~17¬ɑ *ɑ 7ɑ ,Ǵ1ɑ aɑ. YZ w ^w`Zw 67ɑ 7ɑ 7ɑ ,1ɑ aɑ Wn wQwnn _ɑ 6ɑnw 7ɑ ɑQ*c7ɑ  ɑ ɑ ZwSwnjwnw sw 7ɑbɑ 5ɑ 0A. Deriving Best Response Functions p*ɑÑd7ɑ þɑ7ɑ. \kw QYnjw# Wnj wKæ ^w`nkwnw s wnjw ( Vnkuw. „ɑQ7ɑɑDɑɑ*¸*êɑįDƲljȐɑ 6ɑ7 ɑɑnjw_]j7_nkw .A -æ 7ɑ. #`nkw^)`njwnB w " 3 Q p =ɑ ɑ7* gɑ !. Vw nBw. ÏÅKå4æ. U7ɑ7ɑ,17ɑɑ **ɑnkw Dɑɑ1gɑDɑnBw 6ɑ1ɑĽ7ɑ1ɑɑp*ɑ vx7ɑɑ|=ɑnpnB Mw. E#.æ 10. Lecture 4 2013 Winter.

(11) @§ ¸ 7ɑp*ɑKɑ ɑ ɑnvw ɑɑ*7ɑaɑnew Twnonv 1w £1_ɑ ɑSba*ɑ. p*ɑv Ŕ7ɑĆɑ,,ɑp*ɑKđ7ɑɑ7gɑaɑ¿ɑn vwSwnf1w Oǝñɑ7ɑ 7ɑ * bɑ ɑnew Qwnonf 1w Oɑ 7*ɑ 7ɑ bɑ DŅɑp*ɑ K9ɑ ɑ7Ÿɑ ɑ. SǷbǸa*ɑɑ*7ɑb7ɑaɑɑ7ɑɑnfw Twnhne 1w TɑSba*ɑ7ɑ7ɑQɑaɑ. Deriving Nash Equilibrium ɑ77*ɑDɑS7ɑ newP nrnf w. nfwSnhne uw àȬɑX4F § ,17ɑp*ɑvc7ɑɑVň½ɑ`ɑ ɑ7ɑ6ɑɑQ7ɑ SɑDɑ ɑ ,ɑ77*ɑ7ɑ. ngw Y n” XwJ7 njw 6æ U7ɑ Ċɑ6ɑQ*7ɑɑbɑ ~7*ɑ7ɑ Vn_ɑ ɑ Mba*ɑ 6bȏɑ b7ɑaɑ. 7**=ɑ ɑ7_ɑnewQwngw0A nX1w `ɑ7ɑ1ɑ. Y”) n” nXw nwXwK›æ. rb1,ɑDɑnXw7ɑN7ɑ. 11. Lecture 4 2013 Winter.

(12) Best Response Curves (1). *ɑ. 6ɑ. 9ɑ. bgɑ. ɑ. aɑ. ɑ. 7ɑ. LU 12. Lecture 4 2013 Winter 5E9zfŠ \zfY\oŠ0d{ETE"fF{VŠ. IA3).

(13) S".h@")ɑ !4ɑ&ɑ4.&!ɑ ɑ !R4 ɑ4)  -ɑ0ɑ#@ !ɑ}” &5ɑ. Best Response Curves (2) š. Ž§. š_§. šh§. šh§ 13. š_§. š. H§. Lecture 4 2013 Winter 5@:|f,ŠŠ $\V` fEY9Šq;,Š\|fY\oŠ.d|?S?!fA}WŠ?r;ŠX\Y\`\RŠ Y+Š_,i‚,&qŠ(\Va2uEvG\YŠ.

(14) Bertrand or Cournot? “Which model is the best?” or “Why do we need so many models?”  Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.)  Bertrand: when firms can adjust capacities faster than prices, e.g., software.  Cournot: when prices can vary faster than capacities, e.g., wheat, cement.  . 14. Lecture 4 2013 Winter.

(15) Further Remarks The Bertrand and Cournot models are different games, i.e., price competition vs. quality competition.  The unique equilibrium concept (=NE) can explain different market outcomes depending on the situations (markets or industries).  That is, we don’t need different assumptions about firms’ behaviors. ⇒ Once a model is specified, then Nash equilibrium gives us the result of the game.  . 15. Lecture 4 2013 Winter.

(16) Further Exercises  . Does iterated elimination of strictly dominated strategies lead to the Nash equilibrium in the Cournot model?  . If yes, explain the process.. In the spatial competition model, does the Nash equilibrium change if each firm can choose location continuously (i.e., a strategy is real number, not integer)?  Consider the Bertrand model with asymmetric marginal costs and derive the Nash equilibria.  .  . 16. You can assume that the firm with lower marginal cost can take all the market share if the firms charge the same price.. Lecture 4 2013 Winter.

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