Bayesian games and their use in auctions

Adapted from notes by Vincent Conitzer

What is mechanism design?• In mechanism design, we get to design the

game (or mechanism)– e.g. the rules of the auction, marketplace,

election, …• Goal is to obtain good outcomes when agents

behave strategically (game-theoretically)• Mechanism design is often considered part of

game theory• 2007 Nobel Prize in Economics!• Before we get to mechanism design, first we

need to know how to evaluate mechanisms

Which auction generates more revenue?• Each bid depends on

– bidder’s true valuation for the item (utility = valuation - payment),– bidder’s beliefs over what others will bid (→ game theory),– and... the auction mechanism used

• In a first-price auction, it does not make sense to bid your true valuation– Even if you win, your utility will be 0…

• In a second-price auction, it always makes sense to bid your true valuation

0

bid 1: $10

bid 2: $5

bid 3: $1

Are there other auctions that perform better? How do we know when we have found the best one?

0

bid 1: $5

bid 2: $4

bid 3: $1

a likely outcome for the first-price mechanism

a likely outcome for the second-

price mechanism

Collusion in the Vickrey auction

0

b = highest bid among other bidders

• Example: two colluding bidders

price colluder 1 would pay when colluders bid truthfully

v2 = second colluder’s true valuation

v1 = first colluder’s true valuation

price colluder 1 would pay if colluder 2 does not bid

gains to be distributed among colluders

The term “Bayesian”• "Bayesian" refers to the mathematician and

theologian Thomas Bayes (1702–1761), who provided the first mathematical treatment of a non-trivial problem of Bayesian inference.

• Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities

The probability of H and D (joint probability) can be thought of in two ways:

P(H,D) = P(H|D) * P(D)

= P(D|H)*P(H)

Now, driving is not independent from gender. Given someone is a girl, what is theprobability she doesn’t drive? Given someone doesn’t drive, what is the probability she is female?

Bayes theorem: The posterior probability is proportional to the likelihood of the observed data, multiplied by the prior probability. Thus

P(H|D) = P(D|H)*P(H)/P(D)• P(H) is the prior probability of H: the probability that H is

correct before the data D are seen.• P(D|H) is the conditional probability of seeing the data D

given that the hypothesis H is true. This conditional probability is called the likelihood.

• P(D) is the probability of D.• P(H|D) is the posterior probability: the probability that H is

true, given the data and previous state of beliefs about H.

Example• Marie is getting married tomorrow. In recent years, it has

rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 5% of the time. What is the probability that it will rain on the day of Marie's wedding?

• P(rain) = 5/365• P(not raining) = 360/365• P(weatherman predicting rain|rain) = .9• P(weatherman predicting rain|does NOT rain) = .05

What is the probability of rain, given the weatherman predicts rain?

• P( rain | predict rain) =   P(rain) P( predict rain| rain ) / P(predict rain)

P(predict rain) = [P( rain)* P( predict rain| rain ) +

P( no rain)* P( predict rain| no rain ) ]

P( rain | predict rain) =

(0.014)(0.9) / [ (0.014)(0.9) + (0.986)(0.05)

P( rain | predict rain)] = 0.2035

Bayesian games• What if we didn’t even know what game is being

played? (Note that in imperfect information games, we didn’t know prior moves, but we at least knew the payoffs)

• On the exam, the “gift” question – both players knew whether or not a gift had been given, but suppose they didn’t have that knowledge?

• We say the “type” of agent determines the game we are playing

Expected Utility

• Ex post: utility based on all agent’s actual types• Ex interim: agent knows his own type, but not

that of others• Ex ante: the agent does not know anybody’s

type

Bayesian games • In a Bayesian game a player’s utility depends on that player’s type as well

as the actions taken in the game – Notation: θi is player i’s type, drawn according to some distribution from

set of types Θi

– In this example, each player knows/learns its own type, not those of the others, before choosing action

– Pure strategy si is a mapping from Θi to Ai (where Ai is i’s set of actions)– In general players can also receive signals about other players’

utilities. Assume all know the probabilities of being in each game.

2,0 0,2

0,2 2,0

U

D

L R

row playertype 1

row playertype 2

2,2 0,0

0,0 1,1

U

D

L R

2,2 0,3

3,0 1,1

2,1 0,0

0,0 1,2

column player

(.3) (.1)

(.2) (.4)

Induced normal form (based on probabilities – no types are known)

LL LR RL RR

UU 2, 1 1, .7 1, 1.2 0,.9

UD .8, .2 1, 1.1 .4, 1 .6, 1.9

DU 1.5, 1.4 .5, 1.1 1.7, .4 .7, .1

DD .3, .6 .5, 1.5 1.1, 0.2 1.3, 1.1

If we had a NE in this induced normal form, it would be called a Bayes NE.

Induced normal form (based on probabilities given player 1 knows his type is type 1)

LL LR RL RR

UU 2, 0.5 1.5,.75 .5, 2 0, 2.25

UD 2, 0.5 1.5,.75 .5, 2 0, 2.25

DU .75. 1.5 .25, 1.75 2.25, 0 1.75,.25

DD .75. 1.5 .25, 1.75 2.25, 0 1.75,.25

Since player 1 knows he is in top games, in left game ¾ time, right ¼.

Could rewrite with only two rows

LL LR RL RR

U 2, 0.5 1.5,.75 .5, 2 0, 2.25

D .75. 1.5 .25, 1.75 2.25, 0 1.75,.25Since player 1 knows he is in top games, in left game ¾ time, right ¼.

Meaningless to find Nash equilibrium, as payoffs aren’t common knowledge

With common valuations• E.g. bidding on drilling rights for an oil field• Each bidder i has its own geologists who do tests, based on

which the bidder assesses an expected value vi of the field• If you win, it is probably because the other bidders’

geologists’ tests turned out worse, and the oil field is not actually worth as much as you thought– The so-called winner’s curse– So if you have common valuation, what would you bid in a

second price auction?

With common valuations• E.g. bidding on drilling rights for an oil field• Each bidder i has its own geologists who do tests, based on

which the bidder assesses an expected value vi of the field• If you win, it is probably because the other bidders’

geologists’ tests turned out worse, and the oil field is not actually worth as much as you thought– The so-called winner’s curse

• Hence, bidding vi is no longer a dominant strategy in the second-price auction

• In English and Japanese auctions, you can update your valuation based on other agents’ bids, so no longer equivalent to second-price

• In these settings, English (or Japanese) > second-price > first-price/Dutch in terms of revenue

Problem• You and your siblings have been left with

an estate consisting of many items• The value of each item is correlated

(partially common and partially private)• How can you divide the property so that

the division is fair?

Auctions without a seller• faculty from a certain department in a university

may want to decide who get to use a limited number of seminar rooms on a certain day, so that the department as a whole benefits the most;

• a group of housemates who collectively own a car may want to decide who gets to use it on a particular weekend, so that the one who needs it the most gets to use it.

• Agents are self interested and may lie about their valuation.

Caution• We are programmed to think that auctions are

about earning money. It may be that we are only trying to distribute goods fairly.

• In the case of dividing the estate, if there is no cost, people could be greedy or overstate their utility (given an inefficient allocation, right?).

• Children’s book: Instead of charging people for items, everything was given away. The logic was that people who made baseball bats loved to do it, so they were happy to give them away to someone who would enjoy it. Then, they could make more.

• I remember thinking this was a great idea.

• no deficit: it is reasonable to assume that a mechanism shall not subsidize any allocation it comes up with, i.e. the total payments made by the agents to the center must always be non-negative

• a stronger version of the no deficit property called strong budget balance, which says the total payments made by the agents to the center must be zero (useful property if jointly owned)

• strategy proofness means `truthful in dominant strategy'.

VCG mechanism fails to have the strong budget balance property (as the auctioneer makes money), it is in fact impossible to find a mechanism that has this property and at the same time is truthful and efficient in dominant strategy

Solutions?• Redistribute as much of the VCG revenue as

possible back to the agents• Sacrifice truthfulness• Sacrifice efficiency (social optimality)

Vickrey auction without a seller

v( ) = 2 v( ) = 4 v( ) = 3

pays 3(money wasted!)

But if there is no cost, people won’t be honest about their valuation

Can we redistribute the payment?

v( ) = 2 v( ) = 4 v( ) = 3

pays 3receives 1receives 1

receives 1

Idea: give everyone 1/n of the payment

not strategy-proof: A loser (bidding higher) increases his redistribution payment

He gets 1/n of the “value” as determined by the auction- so he benefits by increasing the selling price

Incentive compatible redistribution[Bailey 97, Porter et al. 04, Cavallo 06]

v( ) = 2 v( ) = 4 v( ) = 3

receives 1

Idea: give everyone 1/n of second-highest other bids

Strategy-proofYour redistribution does not depend on your bid;

incentives are the same as in Vickrey

Who pays what?

Incentive compatible redistribution[Bailey 97, Porter et al. 04, Cavallo 06]

v( ) = 2 v( ) = 4 v( ) = 3

pays 3receives 1receives 2/3

receives 2/3

Idea: give everyone 1/n of second-highest other bids

Strategy-proofYour redistribution does not depend on your bid;

incentives are the same as in Vickrey

2/3 wasted (22%)

Bailey-Cavallo mechanism…

• Bids: V1≥V2≥V3≥... ≥Vn≥0• First run Vickrey auction• Payment is V2 by bidder 1• First two bidders receive V3/n• Remaining bidders receive V2/n• Total redistributed: 2V3/n+(n-

2)V2/n• So why can’t we give it all to the

remaining bidders?

R1 = V3/n

R2 = V3/n

R3 = V2/n

R4 = V2/n

...Rn-1=

V2/nRn =

V2/n

Another redistribution mechanism

• Bids: V1≥V2≥V3≥V4≥... ≥Vn≥0• First run Vickrey• Redistribution:

Receive 1/(n-2) * second-highest other bid,

- 2/[(n-2)(n-3)] third-highest other bid

• Total redistributed:

V2-6V4/[(n-2)(n-3)]

R1 = V3/(n-2) - 2/[(n-2)(n-3)]V4

R2 = V3/(n-2) - 2/[(n-2)(n-3)]V4

R3 = V2/(n-2) - 2/[(n-2)(n-3)]V4

R4 = V2/(n-2) - 2/[(n-2)(n-3)]V3

...

Rn-1= V2/(n-2) - 2/[(n-2)(n-3)]V3

Rn = V2/(n-2) - 2/[(n-2)(n-3)]V3

Idea pursued further in Guo & Conitzer 07 / Moulin 07

Key point is their return is never a function of their bid