Complete Shrimp Game Solution

Transcription

1 Complete Shrimp Game Solution Florian Ederer Feruary 7, 207 The inverse demand urve is given y P (Q a ; Q ; Q ) = 00 0:5 (Q a + Q + Q ) The pro t funtion for rm i = fa; ; g is i (Q a ; Q ; Q ) = Q i [P (Q a ; Q ; Q ) 0] = Q i [00 0:5 (Q a + Q + Q ) 0] where we sustituted for the market prie P (Q a ; Q ; Q ). Cooperation Under perfet ooperation, the three shrimpers ehave to maximize joint pro t, whih is equivalent to what a monopolist would do. Let Q = Q a + Q + Q e the monopolist s quantity hoie. So the monopolist maximizes the following pro t funtion: Monopoly (Q) = Q [00 0:5Q 0] = 00Q 0:5Q 2 0Q Di erentiating with respet to Q and setting the resulting expression equal to 0 yields the following rst-order ondition: 00 Q 0 = 0 or 00 Q = 0 whih is just marginal revenue on the left-hand side set equal to marginal ost on the right-hand side of the equation. Solving this equation for Q yields the optimal monopoly quantity Q monopoly whih is Q Monopoly = 90 and pro ts Monopoly = 4050:

2 This ollusive outome an e implemented in any way suh that Q a + Q + Q = 90 There are many ways to do this. The symmetri solution is Q a = Q = Q = 30 suh that i = Q i [00 0:5 (Q a + Q + Q ) 0] = 30 [00 0:5 ( ) 0] = 30 [90 0:5 (90)] = 350 Note, however, that we ould also have non-symmetri ollusion, e.g. Q a = 40 Q = 30 Q = 20 2 Optimal Strategies for (and against) Defetion Assume that there is a symmetri ollusive agreement suh that everyone produes Q i = 30. What would e the optimal defetion strategy of Arnold in this ase? The pro t funtion for shrimper A is given y a (Q a ; Q = 30; Q = 30) = Q a [00 0:5 (Q a ) 0] = 00Q a 0:5 (Q a ) Q a 0Q a Di erentiating with respet to Q a and setting the resulting expression equal to 0 yields the following rstorder ondition: 00 0:5 (2Q a ) 0 = 0 or 00 0:5 ( ) Q a = 0 whih is just marginal revenue of the residual demand urve on the left-hand side set equal to marginal ost on the right-hand side of the equation. Solving this equation for Q a yields the optimal defetion quantity Q Defetion = 60 2

3 and the defetion pro t Defetion = 60 [00 0:5 ( ) 0] = 60 [90 60] = 800 The pro ts for Beatrie and Charlotte who oth hoose 30 are 900. Now, what would happen if shrimper A knew that the other two players were going to defet, that is shrimper B and C would oth hoose 60. In that ase he would solve the same prolem exept the pro t funtion is now a (Q a ; Q = 60; Q = 60) = Q a [00 0:5 (Q a ) 0] Following the same proedure of di erentiating with respet to Q a we now nd that the optimal quantity against defetors is Q Against defetion = 30 and the resulting pro t is Against defetion = Q a [00 0:5 (Q a ) 0] = 30 [90 75] = 450 The pro ts for Beatrie and Charlotte who oth hoose 60 are Nash Equilirium Nash Equilirium is de ned y the set of quantities Q a, Q and Q suh that if Arnold assumes that if Arnold assumes that Beatrie and Charlotte are produing Q and Q, it is optimal for Arnold to hoose Q a. Likewise it must e optimal for Beatrie and Charlotte to produe those quantities given optimal quantity hoies y the other players. Thus, eah shrimper produes optimally given the orretly antiipated ations of rivals. Consider rst Arnold s deision. His pro t funtion is a (Q a ; Q ; Q ) = Q a [P (Q a ; Q ; Q ) 0] = Q a [00 0:5 (Q a + Q + Q ) 0] To nd his reation funtion, that is his optimal response to any given quantities Q and Q hosen y Beatrie and Charlotte, we di erentiate with respet to Q a and set the expression equal to zero. The rstorder ondition is 00 0:5 (2Q a + Q + Q ) 0 = 0 or 00 0:5 (2Q a + Q + Q ) = 0 whih is marginal revenue of the residual demand urve given Q and Q on the left-hand side set equal to 3

4 marginal ost on the right-hand side of the equation. We solve for Q a to otain the est-response quantity for Arnold: a = 90 0:5 (Q + Q ) is Now, onsider Beatrie s deision. She solves exatly the same symmetri prolem sine her pro t funtion a (Q a ; Q ; Q ) = Q [P (Q a ; Q ; Q ) 0] : Repeating the analysis yields Beatrie s est-response funtion = 90 0:5 (Q a + Q ) Next, we repeat the analysis for Charlotte to otain her est-response funtion = 90 0:5 (Q a + Q ) : Now that we have the three reation funtions (3 equations in 3 unknowns), all we need to do is to solve this system of linear equations. One way of doing this (there are more elegant ways of ourse!) is to rst sustitute for Q in the expression for = 90 0:5 fq a + [90 0:5 (Q a + Q )]g Next, solve this equation for Q to otain another est reponse and sustitute this expression for Q into the expressions for and a, re-arrange and sustitute again to nd that the optimal quantities are Q Nash a = Q Nash = Q Nash = 45: A more elegant way of solving this prolem would e to realize that the prolem is symmetri, so in equilirium the three players will e produing the same quantities, that is Q Nash a = Q Nash = Q Nash. Hene, we ould take the est-response funtion of Arnold and just use the symmetry property to solve for the optimal quantity, that is a = 90 0:5 (Q + Q ) = 90 0:5 (Q a + Q a ) = 90 Q a whih gives the same solution Q Nash a = 45: The Nash equilirium pro ts for the three shrimpers are Nash a = Nash = Nash = : 4

5 4 First-Mover Equilirium In one of the rounds we played, Arnold ommitted to prodution rst. If he elieves that his ompetitors will respond y produing optimally onditional on his deision, the unonstrained optimal hoie for Arnold is larger than 45. To see this, onsider rst the prolem faed y Beatrie. From efore we know that Beatrie s est-response funtion is = 90 0:5 (Q a + Q ) : This is the optimal quantity to produe for her for any given level of prodution y Arnold and Charlotte. Similarly, for Charlotte the est-response funtion is = 90 0:5 (Q a + Q ) When Arnold makes his prodution deision he already antiipates the est response of Beatrie and Charlotte. So, the pro t funtion for Arnold is a Q a ; ; = Q a P Qa ; ; = Q a 00 0:5 Qa Now the di erene to the setting efore is that Arnold now knows how Beatrie and Charlotte are going to ehave eause they see his quantity hoie. We an use their est response funtions in Arnold s pro t funtion efore we di erentiate with respet to Q a. In other words, when deiding on the optimal quantity Arnold takes into aount that his quantity hoie will in uene the later quantity hoie y Beatrie and Charlotte. In the previous setting this was not possile sine Beatrie and Charlotte did not oserve Arnold s quantity hoie and so they ould not e in uened y it. We solve the game y akwards indution y rst solving for the Nash equilirium etween Beatrie and Charlotte for a given quantity produed y Arnold. Using the expressions for e symmetri we nd and aove and noting again that Beatrie s and Charlotte s hoies will Q F ollower = Q F ollower = 60 Now that we know how Beatrie and Charlotte will hoose, we an sustitute their est responses as 3 Q a followers into Arnold s pro t funtion to determine his est quantity. In partiular, we have a (Q a ; Q ; Q ) = Q a 00 0:5 = Q a 00 0:5 Q a Q a Q a Q a 0 We now di erentiate with respet to Q a to otain :5 3 Q a = 0 Solving for Q a we otain while the followers hoose Q F ollower Q Leader a = 90 = Q F ollower = 30: 5

6 It is straightforward to ompute the resulting pro ts whih are Leader a = 350 F ollower = F ollower = 450 Aside: Note though that if the leader ould hange the quantity afterwards, they would lower it to 60. 6