Bayesian Nash Equilibrium Homework Help

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  • 1. 1. Nash equilibrium In this game, player 2 knows which game they are playing but player 1 does not. Thus, player 1 has two strategies available (T and B) regardless of which game she is playing and her decision will be based on the expected payo¤s (Left with probability 1 2 and Right with probability 1 2 ). But, player 2 should choose one strategy each game (Left and Right). This game can be summarised in matrix as below. Player 2 A; C A; D B; C B; D Player 1 T 2; (2; 2) 4; (2; 0) 1 2 ; (4; 2) 5 2 ; (4; 0) B 1; (2; 0) 5 2 ; (2; 3) 1 2 ; (1; 0) 2; (1; 3) If player 1 chooses T, player 2 has no incentive to deviate from B to A and no incentive to deviate from C to D. And, if player 2 chooses fB; Cg, player 1 has no incentive to deviate from T to B. ) Pure strategy NE : (T; fB; Cg) 2. Restaurant I own a restaurant and know the worth, but you know its value is evenly distributed between 0 and 1. And, if the restaurant is worth X to me, then it is worth 1:5X to you. De…ne price that you o¤er as p. The person making the o¤er must calculate the expected value of the restaraunt conditional on the seller accepting. The seller only accepts a price of p if X p. Therefore, E[Xjo¤er accepted] = p 2 . For any o¤er of p, either the o¤er is declined or the buyer makes an expected pro…t of 1:5E[Xjo¤er accepted] p = 1:5p 2 p < 0. Therefore, the buyer’s best o¤er is to o¤er p = 0, i.e. not to buy at all. This is an illustration of the winner’s curse. The buyer must internalize that the seller accepting the o¤er conveys bad news; speci…cally, it means the restaraunt is not as valuable as he might have previously thought. BAYESIAN NASH EQUILIBRIUM Our online Tutors are available 24*7 to provide Help with Bayesian Nash Equilibrium Homework/Assignment or a long term Graduate/Undergraduate Bayesian Nash Equilibrium Project. Our Tutors being experienced and proficient in Bayesian Nash Equilibrium sensure to provide high quality Bayesian Nash Equilibrium Homework Help. Upload your Bayesian Nash Equilibrium Assignment at ‘Submit Your Assignment’ button or email it to info@assignmentpedia.com. You can use our ‘Live Chat’ option to schedule an Online Tutoring session with our Bayesian Nash Equilibrium Tutors. http://www.assignmentpedia.com/game-theory-homework-assignment-help.html For further details, visit http://www.assignmentpedia.com/ or email us at info@assignmentpedia.com or call us on +1-520-8371215
  • 2. 3. Gibbons 3.2 Inverse demand P(Q) = a Q where Q = q1 + q2 (Uncertainty) aH : with probability aL : with probability 1 (Asymmetricity) Firm 1 knows whether demand is high or not. Firm 2 does not. Both …rms’total cost Ci(qi) = cqi Firm 1 knows the market demand and wants to maximize its pro…t for each state. Thus, the strategy of …rm 1 is qH 1 (when a = aH) and qL 1 (when a = aL). However, Firm 2 does not know the market demand and wants to maximize its expected pro…t. Thus, the strategy of …rm 2 is q2. We also need to consider that output should be nonnegative. That is, q 2 [0; 1): Firm 1’s problem Max qH 1 (aH qH 1 q2)qH 1 cqH 1 @qH 1 : qH 1 = aH c q2 2 (1) Max qL 1 (aL qL 1 q2)qL 1 cqL 1 @qL 1 : qL 1 = aL c q2 2 (2) Firm 2’s problem Max q2 [(aH qH 1 q2)q2 cq2] + (1 )[(aL qL 1 q2)q2 cq2] @q2 : q2 = (aH qH 1 ) + (1 )(aL qL 1 ) c 2 (3) By using (1), (2) and (3), we can get the Bayesian Nash equilibrium. qH 1 = (3 )aH (1 )aL 2c 6 (4) qL 1 = (2 + )aL aH 2c 6 (5) q2 = aH + (1 )aL c 3 (6) Finally, we will consider the nonnegativity condition. Because qL 1 < qH 1 and qL 1 < q2, it is enough to assume that qL 1 0. Thus, our assumption is that aH + 2c (2 + )aL. ) Bayesian NE : (4), (5) and (6) under aH + 2c (2 + )aL http://www.assignmentpedia.com/game-theory-homework-assignment-help.html For further details, visit http://www.assignmentpedia.com/ or email us at info@assignmentpedia.com or call us on +1-520-8371215
  • 3. 4. Gibbons 3.3 Demand for …rm i qi(pi; pj) = a pi bi pj (Sensitivity) bH : with probability bL : with probability 1 y Each …rm knows its own bi but not its competitor’s Both …rms’cost Zero cost The action spaces for …rm i (or j) : Ai = [0; 1) = R+ (* Price can be any nonnegative real number.) The type spaces for …rm i (or j) : Ti = fbH; bLg The beliefs for …rm i (or j) : pi(bHjbi = bH or bL) = ; pi(bLjbi = bH or bL) = 1 The utility function for …rm i (or j) : Ui(pi; pj; bi; bj) = pi(a pi bi pj) The strategy spaces for …rm i (or j) : [0; 1) [0; 1) = R2 + (* Firm i’s strategy (pi(bH); pi(bL)) 2 R2 +) Firm i’s problem when bi = bH, Max pi(bH ) [a pi(bH) bHpj (bH)]pi(bH) + (1 )[a pi(bH) bHpj (bL)]pi(bH) @pi(bH) : pi (bH) = a bHpj (bH) (1 )bHpj (bL) 2 (7) when bi = bL, Max pi(bL) [a pi(bL) bLpj (bH)]pi(bL) + (1 )[a pi(bL) bLpj (bL)]pi(bL) @pi(bL) : pi (bL) = a bLpj (bH) (1 )bLpj (bL) 2 (8) Firm j’s problem when bj = bH, Max pj (bH ) [a pj(bH) bHpi (bH)]pj(bH) + (1 )[a pj(bH) bHpi (bL)]pj(bH) @pj(bH) : pj (bH) = a bHpi (bH) (1 )bHpi (bL) 2 (9) when bj = bL, Max pj (bL) [a pj(bL) bLpi (bH)]pj(bL) + (1 )[a pj(bL) bLpi (bL)]pj(bL) @pj(bL) : pj (bL) = a bLpi (bH) (1 )bLpi (bL) 2 (10) http://www.assignmentpedia.com/game-theory-homework-assignment-help.html For further details, visit http://www.assignmentpedia.com/ or email us at info@assignmentpedia.com or call us on +1-520-8371215
  • 4. We need (11) and (12) conditions to de…ne a symmetric pure-strategy Bayesian NE. p (bH) = pi (bH) = pj (bH) (11) p (bL) = pi (bL) = pj (bL) (12) By using (7), (8), (9), (10), (11) and (12), we can get (13) and (14). p (bH) = a bHp (bH) (1 )bHp (bL) 2 (13) p (bL) = a bLp (bH) (1 )bLp (bL) 2 (14) By using (13) and (14), we can get (15) and (16). p (bH) = a 2 (1 bH 2 + bH + (1 )bL ) (15) p (bL) = a 2 (1 bL 2 + bH + (1 )bL ) (16) ) Bayesian NE : (15) and (16) http://www.assignmentpedia.com/game-theory-homework-assignment-help.html For further details, visit http://www.assignmentpedia.com/ or email us at info@assignmentpedia.com or call us on +1-520-8371215
  • 5. 5. Nash equilibrium (Bertrand) Market demand Q = 100 P where P is the lowest price o¤ered by a …rm Firm 1’s marginal cost 20 Firm 2’s marginal cost 40 with probability 1 5 70 with probability 4 5 y Firm 2 knows its MC, but …rm 1 does not know …rm 2’s MC. We will consider the discrete price case in this problem. Firm 1’s monopoly price Max P1 (100 P1)P1 20(100 P1) @P1 : Pm 1 = 60 (17) We can get …rm 2’s monopoly price in the same way. Pm 2(MC=40) = 70 (18) Pm 2(MC=70) = 85 (19) Each …rm’s best response is as below (under no uncertainty). Firm 1 (with MC=20) BR1(P2)= 8 >>>>< >>>>: 60 if P2 > 60 P2 0:01 if 20:01 < P2 60 20:01 if P2 = 20:01 x (x 20) if P2 = 20 y (y P2 + 0:01) if P2 19:99 Firm 2 (with MC=40) BR2(P1)= 8 >>>>< >>>>: 70 if P1 > 70 P1 0:01 if 40:01 < P1 70 40:01 if P1 = 40:01 x (x 40) if P1 = 40 y (y P1 + 0:01) if P1 39:99 Firm 2 (with MC=70) BR2(P1)= 8 >>>>< >>>>: 85 if P1 > 85 P1 0:01 if 70:01 < P1 85 70:01 if P1 = 70:01 x (x 70) if P1 = 70 y (y P1 + 0:01) if P1 69:99 There is no undominated equilibrium even when prices are discrete. It cannot be an undominated equilibrium for …rm 1 to choose a price close to $40. At best it receives an expected pro…t of $1; 200. However, if it chooses $60 and …rm 2 plays an undominated strategy (P2 70 when MC = 70) then it receives greater expected pro…ts (at least $1; 280 = $1; 600 4 5 ). If …rm 1 chooses any P1 > 40:01, then when …rm 2 has MC = 40, its best response is P2(MC=40) = P1 :01. However, if …rm 2 chooses P2(MC=40) = P1 :01 20% of the time and P2(MC=70) 70 80% of the time, then …rm 1 does better choosing P1 :02. Therefore, there is not undominated equilibrium. However, the equilibria such that …rm 1 chooses any 20:01 P1 40 and …rm 2 chooses P1 +:01 do work. Although it is an equilibrium, it is not one we like because …rm 2 playing P < MC is not reasonable. http://www.assignmentpedia.com/game-theory-homework-assignment-help.html For further details, visit http://www.assignmentpedia.com/ or email us at info@assignmentpedia.com or call us on +1-520-8371215