1. ECON 330 Lecture 17
Monday, November 25

2. Please submit your homework!

3. Updated class participation records are posted on webpage.
Midterm exam grades (updated) and make-up exam grades are posted on course webpage.
Last chance this week if you want to see your exam.
CASE 248
Today 1 30 – 2 30
Tuesday 1 30 – 2 30

4. Last week’s BIG idea was
Commitment and first-mover advantage

5. Cooperation via repeated interaction
The next big idea is
Cooperation via repeated interaction

6. Today’s lecture: We use a simple example to illustrate the main idea.
Today’s lecture is the most game-theory heavy lecture of the semester.
My apologies for that!

7. The example
This is no Cournot, no Bertrand, no Stackelberg. It is something I made up. (based on the example in Chapter 4 section 3 of the textbook.) But I hope it makes sense! I will distribute a handout that summarizes the model…

8. The example There are 2 firms.
Each firm has three options (strategies):
cooperate,
compete,
super-compete.
Firms choose their strategies at the same time.

9. Describing the payoffs (profits) …
If both firms cooperate, each gets 5 millions. If both firms compete each gets 4 millions. If one competes and the other cooperates, the competing firm gets 6 millions and the cooperating firm gets 3 millions. If both firms “super-compete”, each gets 1 million. The other two strategies get negative profits against super- compete, and super-compete gets small but positive profits against the other two strategies.

10. The definition of Nash equilibrium
In a two player game, a Nash equilibrium is a strategy s1* for player 1 and as strategy s2* for player 2 such that
Player 1 cannot do better by choosing a strategy different from s1*, given that Player 2 sticks to s2*.
Player 2 cannot do better by choosing a strategy different from s2*, given that Player 1 sticks to s1*.

11. The definition of Nash equilibrium
Or, we can say the same thing as follows:
For Player 1 s1* is a best response to s2*.
For Player 2 s2* is a best response to s1*.

12. The handouts now

13. Now…
The payoff matrix

14. Monthly Profits (in millions)

15. First, Let’s explain why … both firms choosing “cooperate” is not a Nash equilibrium. Go to the payoff matrix >

16. Monthly Profits (in millions)

17. Let’s find the Nash equilibrium, or equilibria (plural form)

18. There are 2 Nash equilibria!

19. Summary There are 2 Nash equilibria.
Nash equilibrium #1: both firms choose “compete”.
Nash equilibrium #2: both firms choose “super-compete”.
Nash equilibrium #1 is “good”: Each firm gets 4 mill.
Nash equilibrium #2 is “bad”: Each firm gets 1 mill.
There is no Nash equilibrium in which both firms choose “cooperate”,
and get 5 mill each.

20. Repeated interaction can help firms achieve cooperation
Now, the BIG idea:
Repeated interaction can help firms achieve cooperation

21. Please remember: Cooperation between firms that are supposed to compete means less competitive (therefore, higher) prices. Cooperation between firms (to restrict output/raise prices) is bad news for consumer welfare. So, cooperation in this set up is not something good! It will lower social welfare! In the IO terminology it is called “tacit collusion”.

22. Now, back to this…

23. Suppose the interaction occurs twice: round #1, and round #2
In round 2 the firms can look back to round 1 and choose their actions in round 2 accordingly. They can punish bad behavior or reward good behavior.

24. Consider the following strategy for Firm 2
Round #1: “cooperate”.
Round #2…
“compete” if your rival has “cooperated” in round #1, and
“super-compete” if your rival has NOT “cooperated” in round #1.

25. Suppose you are Firm 1
What will you do against this strategy? What is the best response? Go to the payoff matrix >

26. Firm 2: ROUND 1: cooperate
Firm 2: ROUND 1: cooperate ROUND 2: Compete if rival cooperated in round 1, super-compete if rival didn’t cooperate in round 1
Option 2
Round 1: compete,
Firm 2 will super-compete in round 2,
so best round 2 for you is super-compete
Profits? 6 in round 1, 1 in round, total is 7
Option 1
Round 1: cooperate,
Firm 2 will compete in round 2,
so best round 2 for you is compete
Profits? 5 in round 1, 4 in round 2, total is 9

27. This looks like the beginning of a beautiful Nash equilibrium
This looks like the beginning of a beautiful Nash equilibrium! But first… A quick detour about this “history-dependent” round 2 actions…

28. Now consider this
Firm 2’s strategy is: Round #1: “compete”. Round #2: compete if the rival cooperated in round 1; super- compete if the rival didn’t cooperate in round 1. You are Firm 1. What will you do against this strategy? Please write your answer on a piece of paper!

29. and this…
Firm 2’s strategy is: Round #1: “super-compete”. Round #2: “super-compete”, regardless of round 1 actions of the rival. You are Firm 1. What will you do against this strategy? Please write your answer on a piece of paper!

30. and finally this…
Firm 2’s strategy is: Round #1: “cooperate”. Round #2: cooperate if the rival cooperated in round 1; super- compete if the rival didn’t cooperate in round 1. You are Firm 1. What will you do against this strategy? Please write your answer on a piece of paper!

31. The strategies below are a Nash equilibrium of the twice repeated game
Both firms use the following “history-dependent” strategy:
ROUND #1: Cooperate.
ROUND #2: Compete if the rival cooperated in round 1; super-compete if the rival didn’t cooperate in round 1.
This strategy gives the firms higher profit in round 1 than what can be achieved without repetition.

32. more analysis
Round #2 actions: Whatever happened in round 1, the suggested round 2 actions are a Nash equilibrium for round 2. What does that mean?

33. It cannot be a Nash equilibrium (of the twice repeated interaction) where both firms use the following strategy:
ROUND 1: cooperate
ROUND 2: cooperate if the rival cooperated in round 1; super-compete if the rival didn’t cooperate in round 1.
CAN you explain why not?
Please write your answer on a piece of paper!

34. Which actions will you choose against this strategy?
cooperate in round 1
cooperate in round 2 if the rival has cooperated in round 1; super-compete if the rival has not cooperated in round 1.

35. cooperate in round 1 compete in round 2 if the rival has cooperated in round 1; super- compete if the rival has not cooperated in round 1.
Why is this a Nash equilibrium? The total profits from the equilibrium play is 9 (= 5+4) for both players. The maximum achievable total profit by one firm changing its actions is 7 (6+1). Hence, these strategies are a Nash equilibrium of the (twice) repeated game.

36. The general idea: Repetition can sustain cooperation among selfish and rational participants
The threat of triggering the ‘bad’ equilibrium disciplines firms. In other words, threats and promises influence the future behavior.

37. A few terms and definitions

38. Repeated games
A repeated game is a stage game repeated a given (possibly infinite!) number of times.
What is the stage game then?

39. This is an example for the stage game

40. Repeated games
In a repeated game a strategy is a firm’s complete plan of actions for all possible occurrence in the past.
For a two round strategic interaction a strategy for firm 1 has to indicate what to choose in round 1 and what to choose in round 2 for each possible combinations of actions that could have been chosen by the two firms in round 1.

41. For a three round strategic interaction a strategy for firm 1 has to indicate what to choose in round 1,
and what to choose in round 2 for each possible combinations of actions that could have been chosen by the two firms in round 1,
and what to choose in round 3 for each possible combinations of actions that could have been chosen by the two firms in round 1 and round 2.

42. Because firms can react to other firms’ actions, repeated games allow for equilibrium outcomes that would not be an equilibrium in the stage game without repetition.
Bad news:
If the stage game has a unique Nash Equilibrium then, the finitely many times repeated game has also a unique equilibrium: the Nash Equilibrium played in every round.

43. If you repeat this stage game twice, can you get cooperation in round 1?

44. Summary and a few remarks
Towards the end

45. When the interactions between the firms is repeated a number of times, it opens the potential for a reward/punishment strategy.
“If you cooperate in this round, I will continue to cooperate in the next round.”
“If you do not cooperate in this round, then I will not cooperate in future rounds.”

46. Here are some issues
Are these (threats etc) credible (believable) in our example? Whatever had happened in the first round… when they reach the second round, is it not reasonable for both firms to choose “compete”? – This gives a higher profit to both firms than choosing super- compete [ “Let bygones be bygones.”].

47. Should not the firms ‘renegotiate’ if cooperation “breaks down” in round 1 ? If they do, however, both firms choosing “cooperate” in round 1 cannot be sustained as an equilibrium. Note that this credibility issue (is it believable? Will they really carry out their threat to play the bad equilibrium? etc.,) is at the heart of the matter! Commitment!

48. End of lecture Next lecture:
Application: The infinitely repeated Bertrand model of price competition!