September 5 Exercises: Nash equilibrium and dominant strategy equilibrium: p. 44: 1, 3, 4 Example 12 (Continued) Now assume that in addition to payingc v i when the project is provided, each division is required to pay a constant taxt i R regardless of whether or not the project is provided. 1. a. Verify that honest reporting remains the unique dominant strategy of each division. This changes divisioni s payoff to u i (vi,v i)= vi +v i c t i ifvi +v i c t i otherwise The logic of the above analysis remains the same: honest reporting insures that the division receives the larger of its two possible payoffs for allv i. b. What is the deficit incurred by HQ in equilibrium? It incurs the deficit t 1 +t 2 (v 1 +v 2 c) when the project is provided, and it incurs the deficit t 1 +t 2 when it is not. Whether or not these numbers are positive or negative depends on the value oft 1 +t 2. c. Suppose thatv 1 andv 2 are jointly distributed according to the distributionf. Show thatt 1 andt 2 can be chosen so that the expected deficit of HQ is zero. This is easy: choose anyt 1,t 2 such thatt 1 +t 2 =E F [v 1 +v 2 c]. Example 13 Let s return to the problem in which two divisions make reports to a headquarters HQ concerning the values that they would receive if the HQ provided a research project. In the interest of specificity, letv 1 = v 2 = 20 andc = 30. The total value v 1 +v 2 = 40 therefore exceeds the cost of the project and so the HQ should provide it. HQ, however, does not knowv 1 andv 2. Suppose HQ announces to the divisions that each will pay its proportionate share of the cost of the project, i.e., if the reported valuesv1,v 2 sum to more than the cost 30, then the project is provided and divisioniis charged v i v1 +v 2 v1 30 +v 2 Notice that in this method the HQ always exactly covers the costcof the project in the fees that it charges to the divisions. c= v i 1. What is divisioni s payoff or profit function? 17
v i v i v c=20 v 1 i +v 2 v 30 ifv1+v 2 30 1 +v 2 0 ifv1+v 2<30 2. Does divisionihave a dominant strategy? No the best report of divisioniclearly depends upon what the other division reports. If the other division reportsv i 30, for instance, then divisionican report vi = 0 and receive the project without making any payment. If the other division reportsv i =25, then it is optimal for divisionito reportv i =5. Divisioni s best report clearly depends on the report of the other division. It therefore does not have a dominant strategy. 3. Does honest reporting by each division define a Nash equilibrium? If the other division reportsv i =20 (its true value), then the optimal report of division i isv i =10. No, honest reporting does not define a Nash equilibrium. 4. Does there exist a Nash equilibrium pair of reports(v1,v 2 ) in which the project is provided? (The project should of course be provided because the true values to the divisions sum to more than30, the cost of the project.) If the project is provided in the Nash equilibrium, thenv 1 +v 2 30. Claim 1: We first note that in a Nash equilibrium,v 1+v 2=30; if insteadv 1+v 2>30, then, taking the report of the other division as given, divisioniwould profit by reducing its report tov i =30 v i. Ifv1 +v 2 =30, then divisioni s profit is 20 v i v1 30=20 +v 30 30=20 v v i i. 2 This will be 0 ifvi 20 (we don t want either division to want to stop the project because his payoff is negative). I m led to the following conclusion: any pair(v1,v 2) such thatv1+v 2=30 andv1,v 2 20 defines a Nash equilibrium. So it can be a Nash equilibrium for the two divisions to underreport their true values but still profit from having the project provided. Question: Why would we worry about something like the VCG mechanism when the above system seems simpler and manages to get the correct decision made in Nash equilibrium? How plausible is the Nash equilibrium? The two divisions have to choose reports that sum to30 in a Nash equilibrium can they do this without knowing each other s value? What if they don t know each other s value what if each doesn t know if the project should in fact be provided? 5. Is there a Nash equilibrium in which the project is not provided? We needv1 +v 2 <30, in which case each division has a payoff of0. Givenv i, any vi such thatv 1+v2 <30 produces the same payoff of0for divisioni. If divisioni is to profitably deviate with some other report, it must cause the project to be provided. Divisioni s best report for implementing the project is30 v i. To have a Nash 18
equilibrium(v1,v 2) in which the project is not provided, we therefore need v1 +v 2 <30, and 20 30 v 2 30 0, 30 so that division1doesn t want to switch fromv1 to30 v2 and cause the project to be provided, and 20 30 v 1 30 0, 30 so that division2doesn t want to switch fromv2 to30 v1 and cause the project to be provided. This system reduces to v1+v 2 < 30 20 (30 v 2 ) 0 20+v 2 30 20 (30 v1) 0 v1+20 30 The last two inequalities imply that anyv1,v 2 10 define a Nash equilibrium in which the project is not provided. The moral is that while there may be "good" Nash equilibria in this procedure, there are also "bad" Nash equilibria. There are in fact lots of Nash equilibria, which makes it more difficult to have faith that any particular one will be found or selected by the players. definitions: nodes, subgame, extensive form vs. normal form strategy in a dynamic or extensive form game: A player s strategy specifies his choice at every node assigned to him in the game. This includes nodes that are not actually reached or passed through when the game is played and the actions of the other players are taken into account. It is a "complete contingent plan", chosen before the game is played, that fully specifies how a player acts throughout the game against any possible strategies of the other players. equilibrium path: the sequence of actions that are actually observed when the game is played. The issue of credible threats concerns behavior off the equilibrium path that is not actually observed when the game is played. Return to two examples of the last class: Example 14 Predation Game A component of the chain store paradox, which will be discussed later. E: entrant I: incumbent 19
Draw as normal form: E/I Fight Accomodate Out 0,2 0,2 In -3,-1 2,1 Definition. A Nash equilibrium of an extensive form game is subgame perfect if its strategies define a Nash equilibrium in every subgame of the game. This insures that all behavior in the equilibrium is rational (choosing more over less), even behavior off the equilibrium path. The above game has one subgame perfect Nash equilibrium: the Entrant chooses In and the Incumbent chooses Accomodate. subgame, subgame perfect Nash equilibrium, refinement Example 15 The "Divide the Dollar" game in which the dollar is the interval[0,1] (i.e., it is infinitely divisible) has exactly one subgame perfect Nash equilibrium: 1 proposes x=1, and 2 accepts any offer. Notice that the definition of equilibrium specifies how 2 responds to every possible offer by player 1, not just the one that 1 proposes in equilibrium. This emphasizes a subtlety of a player s strategy in an extensive form game: it specifies how he responds in any situation that could arise, not just those that do arise when the game is played because of the choices of the other players. If the dollar is divisible into pennies or cents, then there are two subgame perfect Nash equilibra: 1:x=1, 2: accept any offer, and 1:x=0.99, 2: accept the offer if and only if1 x 0.99. In both the case in which the dollar is infinitely divisible and the case in which it is divisible into cents, it is credible that player 2 turns down an offer that gives him 0 (2 gets 20
0 if he rejects 1 s offer, and he therefore doesn t suffer from rejecting an offer that gives him 0). If the dollar is infinitely divisible, however, we cannot define a best response of player 1 to this strategy of 2 (there is no "smallest positive amount" in the unit interval that 1 can offer 2). We therefore cannot complete the construction of an equilibrium in this case. If the dollar is divisible into cents, however, and player 2 rejects any offer that doesn t give him at least a penny, then 1 s best response is to propose 0.99 for himself and 0.01 for player 2 (which 2 then accepts). Solving a finite extensive form game "backwards induction", which effectively derives a subgame perfect equilibrium. Example 16 The solution of a game by backwards induction (i.e., the determination of a subgame perfect Nash equilibrium), along with second Nash equilibrium: In the bottom game, player 1 chooses a payoff of 1 over a payoff of 2 at the node furthest to the right. With this action, 2 s best response is to choose left over right at the preceding node, earning him a payoff of 5. This is a Nash equilibrium: given the action of the other player, each player s strategy maximizes his own payoff. It is not subgame perfect, however, because of 1 s choice at this particular node. In finite games of complete and perfect information, subgame perfection is exactly the same as solving the game through backwards induction. Why do we bother with it if it is 21
so obvious? It is a useful idea that has significance as a refinement beyond this particular class of games. It is easiest to introduce, however, in the context of this class of games. 22