MBA lecture 7

1. Let's start by considering a new imperfect-information game in which just one player has a dominant strategy: the Fed / Congress game. This describes the situation in which the Fed will respond to a spendthrift Congress with high interest rates, which Congress dislikes; but Congress would best like to run a deficit while the Fed leaves rates low.

Fed

		Low i	High i
Congress	Balanced Budget	3, 4	1, 3
	Deficit	4, 1	2, 2

The NE here is (Deficit, High).

2. There are different possible ways of thinking about NE philosophically. It's a bit sloppy to say that each player makes the best response given what other players `have done' or `will do' because this introduces a misleading temporal dimension into the concept. Think of it this way instead. Player I makes a conjecture about what another player X might do. Then she checks to see how the other players, including herself, could all best respond together to this strategy. Then she looks back at Player X. Does he now have an alternative strategy that would get him a higher payoff? If the answer is `yes' then Player I shouldn't believe her initial conjecture. Only when Player I has checked every possible conjecture for every player, and eliminated all the ones she shouldn't believe, does she know she's found a NE. Thus the best definition of NE is this one: A set (vector) of strategies is a NE iff (1) each player has correct beliefs about the strategies of the others and (2) the strategy of each is the best for herself, given her beliefs about the strategies of the others.

3. The PD is not a typical game in many respects. One of these respects is that all its rows and columns are either strictly dominated or strictly dominant. In any strategic-form game where this is true, elimination of strictly dominated strategies is guaranteed to yield a unique NE. For many games this condition does not apply, and then our analytic task is less straightforward.

4. Sometimes games have no strictly dominant strategies, but have strictly dominated strategies. Then we can eliminate the latter. When we do that, we get a new matrix. This may cause strictly dominant strategies to emerge, in which case we can find the NE. Here's an example:

Column

		Low	Medium	High
	Top	3, 1	2, 3	10, 2
Row	High	4, 5	3, 0	6, 4
	Low	2, 2	5, 4	12, 3
	Bottom	5, 6	4, 5	9, 7

5. Unfortunately, this doesn't always work. Some games gave no strictly dominant or dominated strategies.

6. However, some games that have no strictly dominant or dominated strategies have weakly dominant or dominated ones. A strategy X for a player i is weakly dominated if i has some other strategy Y such that Y always gets i a payoff at least as good as X – that in every cell where i plays X he gets either the same payoff as when he plays Y, or a worse payoff. We can get an example by slightly changing the previous game as follows:

Column

		Low	Medium	High
	Top	3, 1	2, 3	10, 2
Row	High	4, 5	3, 0	6, 4
	Low	2, 2	5, 4	9, 3
	Bottom	5, 6	5, 5	12, 7

7. One must be careful about eliminating weakly dominated strategies, because in their case the order in which it's done matters. This is because some games have multiple NE. In such cases, elimination of weakly dominated strategies can accidentally remove some NE, and which NE it removes and which ones it leaves depends on the order in which the weakly dominated strategies are eliminated. We’ll come back to this and other issues raised by games that have multiple NE later.

8. Here's a matrix from a previous lecture in which we can eliminate one row, but then we get stuck:

	LL	LR	RL	RR
LL	3,3	3,3	0,5	0,5
LR	3,3	3,3	0,5	0,5
RL	-1,0	4,5	-1,0	4,5
RR	-1,0	5,-1	-1,0	5,-1

12. When you can't eliminate any rows or columns from a matrix, you have to adopt the tedious procedure of locating each best response one by one. If we find a cell that's a best response for all players then we've found a NE. If we find no such cell, then we know the game's only NE is in mixed strategies.

13. In a zero-sum game there's a special tactic for trying to find NE: the minimax method. It's based on the following logic: you know that for any strategy you pick, the other player's best response will be the one that gives you your worst payoff. Therefore, your best response to all her possible strategies must be the one that gives you the highest among these. Meanwhile, she's trying to pick the strategy that yields her highest payoff. You want to hold her to the lowest of these. Then if there's one pair of strategies such that there's nothing she can do to make you worse off, that must be the best either of you can do, and so it must be a NE. Here’s an example, with Player I's worst – minimum – payoff for the corresponding row and Player II's best – maximum – payoff for each column written in at the end:

II

		Left	Middle	Right	Min
	Left	20, 60	40, 40	30, 50	20
I	Middle	60, 20	50, 30	80, 10	50
	Right	10, 80	30, 50	10, 80	10
	Max	60	50	80

• I – chooses maximum among the minimums: maximin
• II – chooses minimum among the maximums: minimax
• NE: where maximin = minimax
• Note you only know this is a NE if you know that the game is zero-sum. How can you verify that?

14. To show how to represent a 3-player game in strategic form, we'll consider three selfish neighbors who want a common garden. Their utility functions are such that for each player i:

6 = i does not contribute, j and k do

5 = i contributes, j and k do

4 = i does not contribute, only one of j or k does

3 = i contributes, only one of j or k does

2 = i does not contribute, neither of j or k does

1 = i contributes, neither of j or k does

Now to build a third dimension into two-dimensional matrices we need to add pages. We'll represent Talia's strategies by making a separate page for each of her two strategies. (If she had three possible strategies, we'd need three pages, and so on for every n. Note that we could have chosen any player as the one for whom we add pages.)

Talia Contributes:

Nina

		C	D
Emily	C	5,5,5	3,6,3
	D	6,3,3	4,4,1

Talia Doesn't Contribute:

Nina

		C	D
Emily	C	3,3,6	1,4,4
	D	4,1,4	2,2,2

The player represented on separate pages is Player III, so her payoffs are listed third in each outcome.

Now when we look for dominant strategies, we need to check both pages. Emily is better off not contributing whatever Nina does on both pages. Therefore Emily's dominant strategy is D, and we can eliminate her C row on both pages. Nina is also better off not contributing whatever Emily does on both pages. Therefore we can eliminate her C column on both pages. For Talia we compare her payoffs on matching cells across the two pages. It turns out that Talia has a dominant strategy D as well; therefore we can eliminate the page on which she plays C. And now there's just one cell left: (D, D) on page 2. Thus this is the NE. (We’ve discovered that this is a 3-person PD. PDs involving more than 2 players are called tragedies of the commons because they lead to situations in which agents ruin shared resources. They are much more common in real life than 2-player PDs.)

15. Some games have multiple NE. If all that players care about is doing the same thing as one another, or exactly the opposite of one another, then we get a game with multiple NE called a pure coordination game. In the matrix below, Harry and Sally want to be at the same coffee shop but don't care which one:

Sally

		Starbucks	L Latte
Harry	Starbucks	1,1	0,0
	L Latte	0,0	1,1

In this situation, there are two NE. If Harry goes to Starbucks then Sally can't do better than going to Starbuck's and vice-versa; if Harry goes to Local Latte then Sally can't do better than going to Local Latte and vice-versa. Here, the NE are desirable for the players. Their chances of finding one of them depend on how much they know about one another. If they know very little about each other, their NE strategies are to each flip a coin. They'll then have a 50% chance of meeting, and that's the best they can do.

16. Now let's change the payoffs:

Sally

		Starbucks	L Latte
Harry	Starbucks	1,1	0,0
	L Latte	0,0	2,2

Here they still care about meeting, but they also prefer Local Latte. Note that (Starbucks, Starbucks) is still a NE, because if both choose it then neither does better switching to the other strategy. But one NE is better (more efficient) then the other, and if they both know the structure of their game they'll find it. This is an example of an assurance game.

Goods that have the property of Local Latte in this example – that is, in which people both prefer them and value coordinating with others on them – create bandwagons, self-reinforcing cascades of consumption. This is why Bill Gates is the world's wealthiest person.

17. Now consider the situation where people want to coordinate but prefer to do so on different outcomes. These have unfortunately become known as `battle of the sexes' games because of the story to which the analysis was first applied. As an example, suppose that Harry and Sally want to go to a movie together, but prefer different films. Here is the matrix:

Sally

		Alien vs Predator	Before Sunset
Harry	Alien vs Predator	2,1	0,0
	Before Sunset	0,0	1,2

Once again we have two NE, since if they coordinate on either movie the person who's seeing their less-preferred film would rather forego their favorite than see it alone. However, they'll get coordination breakdown in two cases: if both parties act selfishly or if both act generously. They coordinate (in pure strategies) only if one is selfish and one is generous.

18. Finally, consider chicken, a coordination game in reverse. Here, the worst outcome for both players happens if they do the same (drive straight into one another), but each player wants to be the only one not to swerve, and would rather that both swerve than that he be the only one to do so:

Dean

		Swerve	Straight
James	Swerve	0,0	-1,1
	Straight	1,-1	-2,-2

The two NE outcomes here are the lower-left and upper-right cells. Players have no means of finding them if they confine themselves to pure strategies.

19. Notice that in these multiple NE games we do not expect to observe NE except in the case of assurance games; in the others, non-NE outcomes are often to be expected. The point of analyzing them is to understand why players who value predictability should try to avoid such games, perhaps by taking actions to turn them into other games.

20. The very first game we talked about in the course, with a pitcher choosing a pitch and the batter deciding whether to swing, is a case of a game with no NE in pure strategies. The river-crossing game is another. In these cases, anyone who uses a pure strategy will find it exploited by the other player, and will want to switch. It's no accident that we find good examples in popular sport; below are tennis players choosing shots. In any sport in which there was a pure-strategy NE, players would always use those strategies and the game would be dull. In professional and Olympic curling (a throwing game played on ice) the rules had to be changed about 20 years ago because players discovered that there was a pure strategy NE. This didn't matter in amateur matches where players couldn't perfectly execute the NE strategies; but it made professional and championships games boring, since there were no decisions until someone physically failed. Anyway, here's the matrix for the tennis players: