The Mother & Child Game

Transcription

1 BUS 4800/4810 Game Theory Lecture Sequential Games and Credible Threats Winter 2008 The Mother & Child Game Child is being BD Moms responds This is a Sequential Game 1

2 Game Tree: This is the EXTENDED form of the game Payoffs (utility) C: C: C: C: M: M: M: M: Introduction to Sequential Games Not all games are played simultaneously. In fact, many strategic situations involve sequential decision-making. Consider the simultaneous game of implementing a new communications system. Both Boeing and irbus benefit by choosing the same system (scale effects, learning curve effects for airlines, etc.) Boeing alpha beta irbus alpha beta (100, 50) (40, 40) (25, 25) (50, 100) Verify there are 2 Nash equilibria: i (alpha, alpha); (beta, beta) 2

3 Simultaneous Choice in Extensive Form (game trees) We can capture the exact same game in extensive form. Note the use of an information set to capture the idea that irbus doesn t know whether its at Node 2 or 3 when they make their choice. In other words, irbus must choose without knowing what Boeing has done; this is what makes it simultaneous play. Node 1 lpha B Beta Information Set Node 2 Node 3 lpha Beta lpha Beta (100,50) (40,40) (25,25) (50,100) Payoffs 2-Stage Sequential Games Now suppose the game is played sequentially, where Boeing goes first. Here, irbus knows what choice Boeing has made; i.e., irbus knows where it is in the game tree (nodes 2 and 3 are now in different information sets). To solve this game we use backward induction. Boeing anticipates what irbus will do at nodes 2 and 3, and then makes its choice knowing what irbus will do in response. 2 nd Stage: t node 2, irbus will choose alpha (50>40) t node 3, irbus will choose beta (100>25) 1 st Stage: Knowing how Stage2 will unfold, Boeing will choose alpha (since 100>50) NE consists of the strategy profile {alpha, alpha}. B alpha beta alpha beta alpha beta (100,50) (40,40) (25,25) (50,100) 3

4 Comments Note that while the simultaneous game has 2 Nash equilibria, the sequential game has only 1 Nash equilibrium. Note also that Boeing has an advantage by virtue of choosing first. However, if irbus made its choice first, irbus would have an advantage, and the equilibrium would be different. In this case, irbus would know that Boeing has an incentive to match technologies. Verify that t the NE with irbus choosing first is {Beta, Beta}. You must not conclude from this that all sequential games have first-mover advantages. They don t. Sometimes its pays to move second (e.g., product imitation, process innovation through reverse engineering, etc.). Credible Threats With Boeing choosing first, irbus has an incentive to influence the actions of Boeing in ways favorable to itself. Specifically, irbus would like Boeing to choose beta since 100>50. So how might irbus get Boeing to choose beta? Suppose irbus announces its plan (in the media) to choose beta no matter what Boeing does, which hopefully gives Boeing an incentive to choose beta as well (since 50>40). Is this a credible threat? t? Clearly not! If Boeing ignores irbus threat and chooses alpha, Boeing knows it s in irbus interest to also adopt alpha. Thus, irbus threat is cheap-talk. 4

5 Credible Threats: n Example Mountain & a village 1000 people will climb Mountain t the Top of the Mountain: Each person wants Exactly One Beer Reservation Price for each is $5 i.e. if Price < 5 Demand = 1000 beer if Price > 5 Demand = 0 Credible Threats: n Example SKIPPY decides to take beer up Mountain to sell to climbers Info: Beer in village costs $1 each It costs $1.50/beer to transport each way ($3/beer round trip) Beer at top costs $2.50 Beer brought back costs $4.00 5

6 Credible Threats: n Example If everything goes according to plan: Skippy sells 1000 beer for $5 Skippy s costs are $2.50 Total Revenue = $5000 Total Cost = $2500 Profit = $2500 Credible Threats: n Example However, MYRTLE also decides to bring 1000 Beer to the top of the mountain Skippy and Myrtle arrive at the same time So do the climbers Questions: How many beer are consumed? How many beer are brought back to village? What is the equilibrium price of BEER on the Mountain? 6

7 Credible Threats: n Example Outcomes (notes): Credible Threats (Boeing/irbus cont.) Credible threats require irbus to restrict its own future actions (i.e., not choosing alpha) by making a binding commitment to beta. binding commitment involves ensuring that irbus will choose beta no matter what Boeing does. Only then will Boeing change their beliefs about what irbus will do. For example, suppose irbus signs a longterm contract with beta company. Contract stipulates that if irbus breaches the contract (by choosing alpha) irbus pays beta company 20 in damages. Now, the payoff for {alpha, alpha} is (100,30), and {beta, alpha) is (25,5). Is irbus threat t of choosing beta credible now? Yes! irbus has an incentive to choose beta no matter what Boeing does, and this commitment is sufficient to change Boeing s beliefs. The NE is now {beta, beta}=(50,100). B alpha beta alpha beta alpha beta (100,50) (40,40) (25,25) (50,100)

8 GME TREE: