MBA 201A Section 6: Game Theory and Review

Overview  Game Theory  Costs  Pricing  Price Discrimination  Long Run vs. Short Run  PS 5

Game Theory  We have seen two types of games:  Simultaneous (i.e. one-shot games) – players must choose their behavior / strategies before knowing what the other player will do  Sequential games – players strategies can be influenced by what other folks do and they make their decision in sequence  How to solve each type of game?  For simultaneous games, creating a payoff matrix is usually the way to go. To solve find the Nash equilibrium (each player’s strategy is the best given what the other players are doing)  For sequential games, create a decision tree and remember to keep track of which player moves at each node. To solve, move backwards from the end of the tree

Review of Costs  Different types of costs:  TC = FC + VC  FC = Costs that do not vary with quantity  VC = Costs that do vary with quantity  MC = cost of one additional unit of production = dTC/dQ  AC = TC/Q  Opportunity cost = value of next best opportunity  What else could you be doing with your resources?  Sunk costs = costs already expended or non-recoverable  Is the cost behind you on the decision tree?

Review of Pricing  Perfectly competitive firm sets P=MC to maximize profits  Since firms are price takers and can sell as many units as they choose at price P, MR = P for any firm.  In short-run equilibrium (without entry) firms can be profitable, if AC MC then a firm should shut down.  Monopolist sets MR=MC to maximize profits  Find MR by taking derivative of R=P(Q)*Q, or if you have a demand curve of the type P=a - b*Q, remember that MR=a - 2b*Q.  Revenues are not maximized at the same production point as profits, unless MC=0 (revenues are maximized where MR=0).

Review of Price Discrimination  Price discrimination allows monopolist to charge different rates to different consumers.  Only meaningful for a monopoly (Why?)  If some kind of PD is profitable, it helps the monopolist vs. single price benchmark.  Ambiguous effect on consumers (some customers previously “shut out” of the market may get to consume).  1st degree: charge each consumer its WTP  Not seen very often in real life  3rd degree: sort customers based on some observable trait where it’s legal to charge different prices (e.g.: student tickets at movies)  2nd degree PD and its relatives: allow customers to self select (versioning, intertemporal PD, quantity discounts)  Give customers the incentive to self-select by making their consumer surplus greatest for the product type you want them to select.

Tips for 2 nd Degree PD Problems  Set up strategies or a “menu of options” and methodically calculate the prices which get customers to do what you want them to do. Pick the option that maximizes profit.  Some options to try: 1.Sell one product, only to high valuation group. 2.Sell one product to everyone (note high valuation group will get rent). 3.Set up a 2nd degree PD scheme.  General rules for setting up 2nd degree PD scheme: 1.Always charge low WTP group its maximum WTP for low quality product. 2.Make sure that high WTP group buys high quality product by giving more than CS from choosing low quality product.

Short Run vs. Long Run  Short run  There may be fixed costs; the number of firms is fixed.  Set MR=MC; exit if P<min AVC  Recall t-shirt example; may want to stay open if P<ATC  Long run  No fixed costs, all costs are variable.  This is definitional; hence “long run” varies by industry.  In long run, P=MC=minAC.  Firms with AC<MC make positive economic profit  enter.  Firms with AC>MC make negative economic profit  exit.

Long Run Equilibrium  Long-run competitive equilibrium  All firms in industry are maximizing profit.  No firm has incentive to enter or exit.  Price of product equates Q s with Q d.  Occurs where economic profits = 0 and MC = minAC  To get the minimum point on an AC curve, you have 2 options: 1.Set MC=AC and solve (since they cross where AC is lowest). 2.Take dAC/dQ and solve for where this equals 0.

 Part (a) Remember, here I write the payoffs in the form (Luxor, Candel). This is equivalent to method in the answer sheet. Just do whatever works for you.  Part (b) Advertise is the dominant strategy for both firms. Why? Imagine you are Luxor and you KNOW that Candel has played No Advertise. What do you do? If you play Advertise you get a payoff of 2.5 vs. 2. So you want to play Advertise (Note: I am underlining the optimal payoffs in the table above to keep track of the strategies). Now let’s say you KNOW Candel has played Advertise. If you play No Advertise, then you get 0.5. However, playing Advertise gets you 1 which is greater. Therefore no matter what Candel does, playing Advertise is a dominant strategy for Luxor. As the game is symmetric, I can do the same for Candel.  Regardless of Candel’s choice, I ALWAYS want to play Advertise. So it’s the dominant strategy (and similarly for the other firm). No Advertise is then a dominated strategy for both firms PS 5 Question #1 (1, 1)(1, 1)(2.5, 0.5) (0.5, 2.5)(2, 2) AdvertiseNo Advertise Advertise No Advertise Candel Luxor

 Part (b) cont’d (Advertise, Advertise) is also the Nash Equilibrium for the game. We can see this two ways: first, it’s the only box with two “underlines” in it. Secondly, if both players have strictly dominant strategies then by definition they should always play them. Which means that this outcome is also a Nash Equilibrium. PS 5 Question #1 (cont’d) (1, 1)(1, 1)(2.5, 0.5) (0.5, 2.5)(2, 2) AdvertiseNo Advertise Advertise No Advertise Candel Luxor

PS 5 Question #1 (1, 1)(1, 1)(2.5, 0.5)(3.5, 0) (0.5, 2.5)(2, 2)(4.5, 0) (0, 3.5)(0, 4.5)(0, 0)  Part (c) Adding a third option means that equilibrium has not changed (Advertise, Advertise), however, Advertise is no longer a dominant strategy. Why? Because, if I am Luxor and I KNOW that Candel is playing No Entry, then I actually want to play No Advertise as its payoff is 4.5 vs So now we have a case where Luxor will want to play something other than Advertise. Is No Entry a dominated strategy? Yes, because there is no situation where it is better to play No Entry. NO MATTER what the other firm plays, you can always switch from No Entry and get a higher payoff (ie greater than the 0 you get from No Entry). The Nash Equilibrium in this game has not changed. Another way to see this – if No Entry is a dominated strategy for both players then you can IGNORE it. If you remove that option from the game then the game just collapses to what we found in part (a), ie a 2x2 matrix. AdvertiseNo AdvertiseNo Entry Advertise No Advertise No Entry Candel Luxor

Another Example (1, 1)(1, 1)(2, 0)(3, 0) (0, 2)(3, 3)(0, 0) (0, 3)(0, 0)(9, 9)  Now let’s change the payoffs a bit. (No Entry, No Entry) now results in payoffs of (9, 9). It could be that if they both do not enter they will look more attractive for a LBO firm who will buy them and merge them together. But the LBO firm is only interested in the combined company and doesn’t want to get involved when one of the companies enters the market and starts spending money.  Are there any dominated strategies? NO – here each firm can play one of the strategies depending on the other firm’s actions. E.g. if Luxor KNOWS that Candel is playing No Entry, then Luxor’s optimal strategy is to play No Entry as well because that gives a payoff of 9 vs. 3 vs. 0.  Are there any dominant strategies? NO – Luxor can play a different strategy depending on what Candel does. i.e. each strategy for Luxor has an “underline” for its payoffs.  What are the Nash Equilibrium(s)? (Advertise, Advertise), (No Advertise, No Advertise), (No Entry, No Entry). So there are three Nash equilibriums in this game. See the boxes that have the two underlines….you should go through solving it yourself. AdvertiseNo AdvertiseNo Entry Advertise No Advertise No Entry Candel Luxor