Game theory

7/17/2019 Game theory

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Perspectives in Mathematics: Notes onGame Theory, part 3

Homework Write a page on one of the problems/questions mentioned in these notes(or something else similar that you have come across in your reading.) Due: Wed Sept30 in class.

If you have had to miss a class, please write two pages. (that way you will get 2points)

Network problems are very important if you a traffic engineer for a road system orfor the internet....

Braess’s paradox Here we have a network of 4 roads in (I) on the left of Fig f:1

0.0.1 ,with an added road in (II) of the right.

Think of this as a game with many players (the number of drivers) each trying tominimize the time they take.

s)II ()I (

BA

C

DD

C

BA

0

45

4545 100x=

100x=

100x=

45100x=

Figure 0.0.1. Network for Braess’s paradox. Each road is labelledwith time in minutes that it takes to traverse it, where x is the numberof cars. f:1

Suppose there are 4000 cars in all.Consider situation (I). If x cars go from A to B via C the time takes is x/100 + 45,

which is the same as the time it would take going via D. Therefore both roads (onethrough C and one through D), are equivalent. The time taken would be least if the cars divide themselves equally between the roads. So the average time taken is2000

100 + 45 = 65. This is a Nash equilibrium, in the sense that if the cars divide equally

no single car could do better by choosing the other route. Note also that when thetotal number of cars is 4000 there is no other Nash Equilibrium: if x < 2000 cars govia C , then it takes longer to go via D , and any single car going through D could takeless time by switching to C .

Now suppose that a wide road is built from C to D , so wide that it takes time 0 to

go from C to D. Then any car arriving at C from A would be better off going via D(at the worst it would take 4000

100 = 40 < 45 mins) rather than going directly along C B.

Similarly any car at A would be better off going to C first, rather than to D. Thus1



2

the Nash equilibrium is for all cars to go from A to C to D to B, with a journey thattakes 80 mins. (which is worse than before!)

This analysis does depend on the total amount of traffic.Note: Apparently there was a situation somewhat like this in Seoul, and the authoritiesmanaged to speed up traffic overall by closing a wide overpass in the middle of the cityto cars.

Theory of auctions There are two kinds of Sealed bid auction. In a First Price

action, players submit their bids to buy an ob ject; these bids are opened. The payerwho makes the highest bid gets the object and pays the price he bid. In a Second

Price action, players submit their bids to buy an object; these bids are opened. Thepayer who makes the highest bid gets the object and pays NOT the price he bid, butthe next highest price.

The second procedure seems counterintuitive. But it leads to a TRUTHFUL auction,i.e. one in which the best strategy is to bid your true value. (This is NOT the beststrategy in a sealed bid first price auction; cf. discussion of ”winner’s curse”) Here weare thinking of a situation/game in which there are repeated independent auctions of asimilar nature (e.g. this is how ad slots on Google are sold), so people need to developstrategies that pay off in the long term.

Note also that the second price auction is much the same as the usual open auction(an “English auction”) with an auctioneer. Here the auctioneer raises the price untilonly one person is left bidding. That person gets the object, but pays essentially (i.e. just a little above) what the second highest bidder offered. In these terms the firstprice auction is a like a “Dutch auction” (e.g. for tulip bulbs) in which the auctioneerstarts with a very high price, gradually lowers it and sells the object to the first personwho bids.

Suppose we have lots of players Ai bidding for some object that they value at vi, but

their bid is bi (presumably ≤

vi). The next result shows what is meant by a truthful

auction.

Theorem: In a sealed bid second price auction, it is a dominant strategy for each

player to choose bi = vi.

To see this, we need to show that if all other players play as before, player i cannotdo better by playing something other than bi = vi. The payoff to player i is

P i =

0 if he/she does not get itvi − b j if he/she gets it and b j is the next highest bid.

Suppose you as player i bid b

i > vi. If you don’t win with b

i then you wouldn’t winwith betting vi and payoff is zero in both cases. If you win with b

i and if you bet vi

then your payoff P i remains the same. So suppose you win with b

i but do not win

if you bet vi. Then the second bid b j ≥ vi. If b j = vi then your new payoff is zero.And it was zero before, because even if you won, the payoff was vi − b j = 0. So again,no change. On the other hand if b j > vi then your payoff when you win with b

i is



3

NEGATIVE, while before you would have had payoff 0 (because you’d wouldn’t havegotten the object.) So this is strictly worse!

So now suppose you as player i bid b

i < bi = vi. Then you have less chance of winning, and if you do win, your payoff is the same as before. So again this strategy isstrictly worse.

Chapter 9 discusses many other aspects of auctions. e.g. what is the best strategyin a sealed bid, first price auction?

Questions

1. Look at the network (II) for Braess’s paradox and find Nash equilibria when thereare 4500, 5000, 6000, 7000 cars. How does the total time taken by all cars (a measureof “social value”) compare with that for network (I)? What do you notice? Are theresituations when network (II) is better?

2. re auctions: there are many interesting questions on p 242. Here is one:How does the number of bidders in a second price sealed bid auction affect how much

the seller can expect to receive? Say there are two bidders with independent private

values vi for the object that are either 1 or 3, with probability 1

2 in each case. If thereis a tie at bid x for the highest bid, then the winner is selected at random and pays x.

(a) Show that the seller’s expected revenue is 6

4.

(b) What happens if in this situation there are three bidders (again each with vi = 1or 3 with probability 1

2.)

(c) Briefly explain why the change in the number of bidders affects the seller’s ex-pected revenue.

5. Question 3 on p 242 is a similar, slightly more elaborate version of this. Question 6asks what happens if there is collusion in a sealed bid, second price auction. Do one of these questions.

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