1. Simultaneous games with continuous strategies Suppose two players have to choose a number between 0 and 100. They can choose any real number (i.e. any decimal). They have continuous strategies If they choose the same number than player 1 wins $1000. Otherwise player 1 gets nothing. If the numbers sum to 100 then player 2 wins $1000. Otherwise player 2 gets nothing. How do we model this game?

2. Best response function for player 1 If player 2 plays 33, then player 1’s ‘best response’ is to play 33 If player 2 plays 74.5 then player 1’s ‘best response’ is to play 74.5 … and so on. Player 1’s best response function tells us, for every possible action by player 2, what is the best response of player 1

3. Best response function for player 1 Player 1 Player 2 0 100 The ‘45 degree line’ gives the best responses for player 1. If player 2 chooses a certain number, player 1’s best response is to choose that same number

4. Best response function for player 2 If player 1 plays 33, then player 2’s ‘best response’ is to play 67 so that the numbers add to 100 If player 1 plays 74.5 then player 2’s ‘best response’ is to play 25.5 so that the numbers add to 100 … and so on. Player 2’s best response function tells us, for every possible action by player 1, what is the best response of player 2

5. Best response function for player 2 Player 1 Player 2 0 100 The line with slope = -1 gives the best responses for player 2. If player 1 chooses a certain number, player 1’s best response is to choose 100 minus that number

6. Nash equilibrium To find the Nash equilibrium, put the best response functions together A Nash equilibrium is a ‘mutual best response’. So a Nash equilibrium occurs where the best response functions cross

7. Nash equilibrium Player 1 Player 2 0 100 The Nash equilibrium is where the best response functions cross. Here it is where both players choose 50. So each player simultaneously and independently choosing ‘50’ is a Nash equilibrium. 50

8. Solving games with continuous strategies For each player find their best response function Where the best response functions cross, we have a Nash equilibrium

9. Oligopoly games We now want to apply simultaneous games to the behaviour of a small number of firms (an oligopoly) Suppose there are just two firms (duopoly) They produce identical goods. But each firm must decide how much it will produce before it goes to sell the product For example, firms must simultaneously choose plant size but then given plant size, they will operate at capacity

10. ‘Cournot’ competition in quantities Firms simultaneously choose quantities The market price then adjusts so that the total quantity produced is sold Each firm sets its quantity to maximise profits Note that each firm has a continuous set of strategies So we need to calculate each firm’s best response function (or reaction function)

11. ‘Firm one’s best response function Output of firm 2 Profit maximising quantity for firm 1 This is the diagram we need to fill in for firm 1. GIVEN a level of output for firm 2, what is firm 1’s best response?

12. Firm one’s best response function Quantity $ Market demand Marginal revenue Marginal cost QMQM PMPM If firm 2 produces nothing, then the profit maximising best response for firm 1 is to produce the monopoly quantity

13. Firm one’s best response function Quantity $ Market demand Marginal cost QMQM PMPM If firm 2 produces and sells a positive quantity then this reduces the ‘residual’ demand for firm 1. Firm 2’s output Firm 1’s demand curve given firm 2’s sales

14. Firm one’s best response function Quantity $ Market demand Marginal cost QMQM PMPM Firm 2’s output Marginal revenue GIVEN firm 2’s output

15. Firm one’s best response function Quantity $ Marginal cost QMQM PMPM So if firm 2 produces more, the best response for firm 1 is to lower output. Note: firm 1 lowers output by less than firm 2’s increase so overall market price falls. Firm 2’s output New optimal quantity New lower market price

16. Firm one’s best response function Quantity $ Market demand Marginal cost PMPM And if firm 2 produces ‘enough’, firm 1’s best response is to produce nothing Firm 2’s output Firm 1’s demand curve given firm 2’s sales

17. Firm one’s best response function Output of firm 2 Profit maximising quantity for firm 1 So we can plot firm 1’s best response function. It starts at the monopoly quantity then falls with a slope less than 1. QmQm Firm 1’s best response function

18. ‘Cournot’ competition in quantities Q2Q2 Q1Q1 We can do the same for firm 2. QmQm Firm 1’s best response function QmQm Firm 2’s best response function

19. ‘Cournot’ competition in quantities Q2Q2 Q1Q1 And get the Cournot equilibrium. QmQm Firm 1’s best response function QmQm Firm 2’s best response function Qc2Qc2 Qc1Qc1

20. Symmetric firms and ‘Cournot’ competition Q2Q2 Q1Q1 If the two firms are identical then Q c 1 = Q c 2. Also total output exceeds the monopoly quantity. QmQm Firm 1’s best response function QmQm Firm 2’s best response function Qc2Qc2 Qc1Qc1 45 degree line (Q m )/2

21. If firm 1’s costs fall then its best response function moves out Suppose firm 1’s marginal costs fall but not firm 2’s Then for any given output of firm 2, firm 1’s profit maximising output will rise So firm 1’s best response function shifts ‘out’ if firm 1’s costs fall

22. If firm 1’s costs fall then its best response function moves out Q2Q2 Q1Q1 Q m (original costs) Firm 1’s new best response function QmQm Firm 2’s best response function Q m (new costs)

23. Firm 1’s costs fall Q2Q2 Q1Q1 Original Q c 2 Original Q c 1 New Q c 1 New Q c 2 So if firm 1’s costs fall, total output rises. Firm 1 produces more in equilibrium and the other firm produces less. Firm 1 makes more profit as (a) its costs are lower and (b) its competitor produces less. Firm 2 makes less profit as (a) total production rises and (b) it produces less. Consumers gain because price falls.

24. Strategic Substitutes Output (or capacity) here is a strategic variable Note that the output that is best for one firm falls as the other firm’s capacity increases For this reason, we call these type of strategic variables ‘strategic substitutes’.

25. Summary For games with continuous strategies, we model the game by looking at best response functions The Cournot competition game has firms simultaneously setting output The firms produce more than monopoly in total (but less than perfect competition) We can capture the strategic effects of a change in costs for one firm

26. ‘Bertrand’ competition in prices Two firms simultaneously choose prices Consumers then decide which firm to buy from Each firm sets its own price to maximise profits As with Cournot competition each firm has a continuous set of strategies Here we will consider the case where the two firms produce imperfect substitutes

27. ‘Bertrand’ competition in prices Here we will consider the case where the two firms produce imperfect substitutes The demand for firm 1 will increase at any price for firm 1 if the price of firm 2 rises. The demand for firm 1 will decrease at any price for firm 1 if the price of firm 2 falls. And vice-versa for firm 2 We need to find the best response functions (or reaction curves)

28. ‘Firm one’s best response function Price of firm 2 Profit maximising price for firm 1 This is the diagram we need to fill in for firm 1. GIVEN a particular price set by firm 2, what is firm 1’s best response?

29. Firm one’s best response function Quantity sold by firm 1 $ Firm 1’s demand Marginal revenue Marginal cost Q1Q1 P1*P1* Firm 1’s demand depends on the price set by firm 2. Here is firm 1’s demand and profit maximising price for a particular value of P 2, the price set by firm 2.

30. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Q1Q1 P1*P1* If firm 2’s price falls, demand for firm 1 falls.

31. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Original P 1 * And as a result, the profit maximising price for firm 1 also falls. New marginal revenue New P 1 *

32. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Q1Q1 And if firm 2 sets a ridiculously low price (e.g. gives its product away) then firm 1’s demand will be very low. Original P 1 *

33. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Q1Q1 In this case, firm 1’s profit maximising price is also very low – but is still positive. New P 1 * Original P 1 * New marginal revenue

34. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Q1Q1 Original P 1 * Of course, if firm 2’s price rises, then demand for firm 1 rises.

35. Firm one’s best response function Quantity sold by firm 1 $ Marginal cost Q1Q1 Original P 1 * New marginal revenue New P 1 * And in this case firm 1’s profit maximising price rises.

36. Firm one’s best response function So if firm 2’s price goes up, the profit maximising price for firm 1 goes up And if firm 2’s price goes down, the profit maximising price for firm 1 goes down So prices are ‘strategic complements’ – they move in the same direction

37. Firm one’s best response function Price of firm 2 Profit maximising price for firm 1 We can plot firm 1’s best response function. It starts at a positive price and slopes up. In general, it also has a slope less than 1 for imperfect substitutes. Firm 1’s best response function

38. ‘Bertrand’ competition in prices P2P2 P1P1 We can do the same for firm 2. Firm 1’s best response function Firm 2’s best response function

39. ‘Bertrand’ competition in prices P2P2 P1P1 And get the Bertrand price equilibrium Firm 1’s best response function Firm 2’s best response function PB1PB1 PB2PB2

40. If firm 1’s marginal cost falls … Suppose firm 1’s marginal cost falls but not firm 2’s Then for any given price of firm 2, firm 1’s profit maximising price will fall So firm 1’s best response function shifts ‘down’ if firm 1’s marginal costs fall

41. If firm 1’s marginal cost falls … P2P2 P1P1 Firm 1’s new best response function with lower marginal cost Firm 2’s best response function PB1PB1 PB2PB2

42. Firm 1’s costs fall P2P2 P1P1 P B 1 (original costs) P B 2 (original costs) P B 1 (new costs) P B 2 (new costs) If firm 1’s costs fall, both firms lower their prices. Firm 2 makes less profit as its demand has fallen. Firm 1’s profit rises as its marginal costs fall as it is cheaper to make its product. But this benefit is at least partially offset by firm 2 dropping its price.

43. Strategic analysis of a drop in firm 1’s marginal costs Suppose firm 1 can invest in new equipment (a fixed cost) that will reduce its marginal cost. Under both Cournot and Bertrand consumers win because prices drop Under both Cournot and Bertrand, firm 2 loses as firm 1 becomes a stronger competitor. Firm 2 ends up with lower profits and a lower price. In both cases firm 1 gains because its marginal costs drop BUT – the strategic effect differs

44. Strategic analysis of a drop in firm 1’s marginal costs Suppose firm 1 can invest in new equipment (a fixed cost) that will reduce its marginal cost. Under Cournot, as firm 1 expands its output, firm 2 ‘backs off’ and gives up market share to firm 1. Under Bertrand, firm 2 responds to firm 1’s increased competitiveness by dropping its price to try and retain its customers So the strategic effect helps firm 1 in Cournot competition and hurts firm 1 in Bertrand competition

45. Which is correct? They both are in the appropriate situations To analyse firm behaviour (or any other strategic situations) you need to study the ‘game’ carefully. ‘Small’ changes in strategic interaction can lead to ‘big’ differences in outcomes (Of course this is why game theory and industrial economics is interesting)

46. Summary We can capture simple strategic interaction between small numbers of firms (oligopoly) using the Cournot or Bertrand models Sometimes these models lead to different predictions – so use wisely More generally, when considering strategy, we need to carefully analyse the real world – there are no simple ‘rules’ that always apply.

47. Reminder: ‘Cournot’ competition in quantities Q2Q2 Q1Q1 Case of two firms QmQm Firm 1’s best response function QmQm Firm 2’s best response function Q1cQ1c Q2cQ2c

48. Strategy for quantity competition If the other firm decreases its output then you can raise your output and increase your profits. You can do this if you lower your rival’s best response function. For example if you push up your rival’s costs If you can commit to increase your output beyond the equilibrium output then your rival will respond by lowering its output This raises your profit so long as you do not increase your output ‘too much’

49. Reminder: ‘Bertrand’ differentiated goods price competition P1P1 P2P2 Firm one’s best response function Firm two’s best response function P1BP1B P2BP2B The case of two firms

50. Strategy for price competition If the other firm increases its price then you can raise your price and increase your profits You can do this by pushing out your rival’s best response function. For example if you push up your rival’s costs If you can commit to increase your price beyond the equilibrium price then your rival will respond by also increasing its price This raises your profit so long as you do not increase your price by ‘too much’

51. So… Strategy is all about changing either your best response function or your rival’s best response function It is about changing the game that you are playing The basic strategic game involves your firm ‘doing something’ before competition between your firm and your rivals

52. The basic strategy game Your firm Choose strategic variable Both firms Market interaction: e.g. Cournot or Bertrand competition

53. But what strategic variables will work? For a strategic variable to ‘work’ it must be a credible commitment. In other words, choosing an action before competition must affect the actual competition game It is not enough to merely threaten and bluster – that is simply cheap talk

54. Cheap talk in the prisoners’ dilemma -10, -100, -30 -30, 0-1, -1 Confess Don’t Confess Ned Kelly Ned Make promise

55. Ned promises to ‘not confess’. Should Kelly believe him and play ‘don’t confess’? No – because it pays Ned to cheat. -10, -100, -30 -30, 0-1, -1 Confess Don’t Confess Ned Kelly Ned Make promise

56. Cheap talk Just promising to do something or threatening to do something that is not in your own interest is cheap talk. To have an effect a strategy must have two requirements It must be credible – in other words it must be something you have done and cannot undo easily It must change the payoffs in the game so that your rival changes its behaviour in a way that benefits you over all.

57. Example – commitment to a high price P1P1 P2P2 Firm one’s original best response function Firm two’s best response function P1BP1B P2BP2B Suppose firm one can commit not to lower its price below P1* P1*P1*

58. Example – commitment to a high price P1P1 P2P2 Firm two’s best response function Firm one’s new best response function P1* This commitment by firm 1 leads firm 2 to also raise its price – so both firms make more profit

59. Example – commitment to a high price The commitment to a high price is a ‘soft’ strategy – it raises your rival’s profit as well as your profit So it changes the game. But how do we make it credible? Cannot just ‘promise to raise price’ Can offer a ‘most favoured customer’ agreement to clients at high price today. This means that if you lower your price in the future, you have to rebate money to all your old customers. It hurts you to drop your price tomorrow – so it is a credible commitment to keep a high price in the future.

60. Example – commitment to a high capacity Q2Q2 Q1Q1 QmQm Firm 1’s best response function QmQm Firm 2’s best response function Q1eQ1e Q2eQ2e Q1*Q1* Suppose that firm 1 can commit to produce at least Q1*

61. Example – commitment to a high quantity Q2Q2 Q1Q1 QmQm Firm 1’s new best response function QmQm Firm 2’s best response function Q1*Q1* This commitment makes firm 2 reduce its output, raising profit for firm 1 and lowering profit for firm 2.

62. Example – commitment to a high quantity This is a ‘tough’ strategy – it raises your profit but reduces your rival’s profit So it changes the game. But how do we make it credible? By building your plant before your rival and By committing to a large capacity and By making it difficult to reverse this choice (i.e. difficult to lower capacity and if it is difficult to run your plant below capacity, or if it is cheap to produce output up to plant capacity Then you have a credible commitment to produce a high level of output. If your rival observes this commitment then your rival will reduce its plant size, raising your expected profits under competition. This is an example of a ‘first mover advantage’

63. Other examples Cortes and burning ships The ‘pub fight’ from tutorials International treaties (e.g. the WTO and protectionism) Mutually Assured Destruction (MAD) and the cold war Reputation (one ongoing player) ‘Rational’ irrationality