R.E.Marks 1998 Lecture 9-1 R.E.Marks 1998 Lecture 9-2 If Alpha preempts Beta, by making its capacity decision before Beta does, then use the game tree: 3. Strategic Moves 3.1 Game Trees and Subgame Perfection Alpha What if one player moves ®rst? Use a game tree, in which the players, their actions, and the timing of their actions are L S DNE explicit. Allow three choices for each of the two players, Alpha and Beta: Do Not Expand (DNE), Small, and Large expansions. Beta Beta Beta The payoff matrix for simultaneous moves is: The Capacity Game L S DNE L S DNE L S DNE Beta DNE Small Large ________________________________ 0 12 18 8 16 20 9 15 18 L L L L 0 8 9 12 16 15 18 20 18 DNE $18, $18 $15, $20 $9, $18 _L_______________________________L L L L L L L Figure 1. Game Tree, Payoffs: Alpha's, Beta's Alpha L L L L Small________________________________ $20, $15 $16, $16 $8, $12 L L L L Use subgame perfect Nash equilibrium, in which L L L L each player chooses the best action for itself at LargeL $18, $9L $12, $8L $0, $0 L _L_______________________________L L L each node it might reach, and assumes similar behaviour on the part of the other. TABLE 1. The payoff matrix (Alpha, Beta) R.E.Marks 1998 Lecture 9-3 R.E.Marks 1998 Lecture 9-4 3.1.1 Backward Induction 3.2 Threats and Credible Threats, or Why Commitment Is Important (See Bierman & Fernandez, Ch. 5) With complete information (all know what each Two ®rms, Able and Baker, are competing in an has done), solve this by backward induction: oligopolistic industry (an industry with few sellers who are engaged in a strategic ªdanceº). 1. From the end (®nal payoffs), go up the tree to the ®rst parent decision nodes. Able, the dominant ®rm, is contemplating its capacity strategy, with two options: 2. Identify the best (i.e. the highest payoff) decision for the deciding player at each node. · ªAggressive,º a large and rapid increase in The choice at each node is part of the player's capacity aimed at increasing its market share, optimal strategy. and 3. ªPruneº all branches from the decision node · ªSoft,º no change in the ®rm's capacity. in 2. Put payoffs at new end = best decision's Baker, a smaller competitor, faces a similar choice. payoffs The table shows the NPV (net present value) 4. Do higher decision nodes remain? associated with each combination of strategies: If ªnoº, then ®nish. 5. If ªyesº, then go to step 1. Baker 6. For each player, the collection of best Aggressive Soft decisions at each decision node of that player ___________________________ → L L L best strategies of that player. Aggressive 12.5, 4.5 16.5, 5 _L__________________________L L Able L L L The simultaneous game (payoff matrix) L L L equilibrium is Alpha: Small; Beta: Small. Soft_L__________________________ 15, 6.5L 18, 6 L The sequential game (game tree) equilibrium is Alpha: Large; Beta: Do Not Expand. TABLE 2. Simultaneous Payoffs (Able, Baker) In the sequential game, Alpha's capacity choice has commitment value: gives Alpha (in this case) ®rst-mover advantage. R.E.Marks 1998 Lecture 9-5 R.E.Marks 1998 Lecture 9-6 There is a unique Nash equilibrium: Able chooses Soft and Baker chooses Aggressive, to give a payoff A to Able of 15. But from Able's point of view, this combination is A S not as good as if both Able and Baker chose Soft → Able's payoff of 18. But without Baker's cooperation, this outcome will B B not be reached. What if Able committed to choose Aggressive whatever Baker chose? If this were credible, A S A S then Baker would choose Soft (for a higher payoff of 5, over 4.5), which in turn would give Able a payoff of 16.5. 12.5 16.5 15 18 4.5 5 6.5 6 How to commit to Aggressive on Able's part? Figure 2. Sequential Payoffs (Able, Baker) It's not enough to announce it or even to promise1 it: not a credible move, since Baker knows that {Able: Aggressive, Baker: Soft} is a subgame Soft is a dominant strategy for Able: no matter perfect Nash equilibrium of the sequential game. what Baker does, Able's payoff is higher if it's Soft. Able may be able to credibly commit by One way is for Able to make a preemptive move, demonstrating that it was rewarding its managers by accelerating its decision process and on market share rather than the NPV pro®t of the aggressively expanding its capacity before Baker payoffs: more pro®table for the managers to go for decides what to do: turns a simultaneous capacity aggressively, even if the company's payoff interaction into a sequential game: appears lower. Paradoxically, Able's position is strengthened if it can reduce its options and tie itself to Aggressive. _________ In¯exibility can have value: strategic 1. Talk is cheap ... because supply exceeds demand. commitments that limit choices can actual improve one's position. R.E.Marks 1998 Lecture 9-7 R.E.Marks 1998 Lecture 9-8 How? 3.2.1 Subgame Perfect Equilibria By altering one's rivals' expectations of about how Nash Equilibria from non-credible threats are poor one will compete, and so altering their decisions. predictors of behaviour. By committing to what seems an inferior decision A subgame: is a smaller game within a larger (Aggression), Able alters Baker's expectations and game with two special properties: its action, to Able's advantage. 1. once players begin playing the subgame, they Commitments must be credible and communicated do so for the rest of the game; and understandable to be of value. 2. the players all know when they are playing · By their nature, strategic commitments the subgame. (threats or promises) are intended to change others' expectations and behaviour; others The subgame's subroot node: the initial node: the must wonder whether the committed player subgame consists of the subroot and all its mightn't fall back on the uncommitted best successors Ð property 1. action: it's nothing but a bluff. If every information set that contains a decision · The movie Dr Strangelove describes a Russian node of the subgame does not contain decision commitment Ð The Doomsday Machine Ð to nodes that are not part of the subgame Ð property respond to any incursion into Soviet airspace 2. with an attack of nuclear missiles on the U.S. The subgame preserves the original game's: Unfortunately, the Russian have overlooked telling the Americans about it ... · set of players, · The rivals' managers must understand the · order of play, implications for their own ®rms' payoffs of · set of possible actions, and Able's ability to price low with its excess capacity. · information sets. To be truly credible, the commitment must be Rational behaviour in the full game should be irreversible: very costly to stop or reverse. rational in the subgame. Non-credible threats are ignored. R.E.Marks 1998 Lecture 9-9 R.E.Marks 1998 Lecture 9-10 Defn: A strategy pro®le is a subgame-perfect 3.2.2 An example: side payments equilibrium2 (SGPE) of a game G if this strategy A sequential game, in which there may be ªside pro®le is also a N.E. for every subgame of G. paymentsº from one player to another: With perfect information (singleton information C sets), the SGPE = those from backwards induction (B.I.). W E B.I. eliminates non-credible threats, so a N.E. ⇔ a SGPE, with perfect information. R R The real power of SGPE occurs when there is not perfect information Ð when players are not always aware of what their opponent did N S N S (modelled by multi-node information sets). So long as there is perfect information, backwards 1,12,±2 ±2,2 0,0 induction results in Nash equilibria that are S.G.P. Side Payments 1 (R,C) The second mover, R, has the following option: · Before his move, he can promise or threaten to reduce his payoff to any positive number or zero by adding the same amount to C's payoff. · This can only happen at one ®nal outcome (only one of NE, NW, SE, or SW). · This is a promise (or threat) of a side payment _________ What would happen without the promise of a side payment? 2. Reinhart Selten received the 1994 Nobel Prize in Economics for his development of this concept. What would happen with such a promise? R.E.Marks 1998 Lecture 9-11 R.E.Marks 1998 Lecture 9-12 3.2.3 Another example: side payments 3.3 A Menu of Strategic Moves A second game tree with side payments possible: 3.3.1 Threats and Promises · An unconditional move may give a strategic C advantage to a player able to seize the initiative and move ®rst. W E · Possible for a second mover to gain similar strategic advantage by commitment to a response rule: ªIf you do/don't act like this, then 4,2 R I'll do/not act like that.º The rule must be in place and clearly communicated beforehand. N S · Two sorts of response rules: threats and promises. 1,1 2,3 · A threat is a response rule that punishes others who fail to cooperate with you. Side Payments 2 (R,C) Ð Compellent threats to induce action (hijacker).