1. Bayesian games and their use in auctions
Adapted from notes by Vincent Conitzer

2. 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

3. 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
bid 1: $10
a likely outcome for the first-price mechanism
a likely outcome for the second-price mechanism
bid 1: $5
bid 2: $5
bid 2: $4
bid 3: $1
bid 3: $1
Are there other auctions that perform better? How do we know when we have found the best one?

4. Collusion in the Vickrey auction
Example: two colluding bidders
v1 = first colluder’s true valuation
v2 = second colluder’s true valuation
price colluder 1 would pay when colluders bid truthfully
gains to be distributed among colluders
b = highest bid among other bidders
price colluder 1 would pay if colluder 2 does not bid

5. 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

6. 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)

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

10. 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.

11. 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

12. 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)] =

13. 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

14. 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

15. Bayesian games L R row player type 1 U D L R U row player type 2 D
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. L R
column player
row player
type 1 U
2,0
0,2
2,2
0,3
3,0
1,1 D
(.3)
(.1) L R U
row player
type 2
2,2
0,0
1,1
2,1
0,0
1,2
(.2)
(.4) D

16. 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.

17. 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 DU
.25, 1.75
2.25, 0
1.75,.25 DD
Since player 1 knows he is in top games, in left game ¾ time, right ¼.

18. Could rewrite with only two rowsLL LR RL RR U
2, 0.5
1.5,.75
.5, 2
0, 2.25 D
.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.
Meaningless to find Nash equilibrium,
as payoffs aren’t common knowledge

19. Bayes-Nash equilibrium
A profile of strategies is a Bayes-Nash equilibrium if it is a Nash equilibrium for the normal form of the game
Minor caveat: each type should have >0 probability
Alternative definition: for every i, for every type θi, for every alternative action ai, we must have:
Σθ-i P(θ-i) ui(θi, σi(θi), σ-i(θ-i)) ≥
Σθ-i P(θ-i) ui(θi, ai, σ-i(θ-i))

20. First-price sealed-bid auction BNE
Suppose every bidder (independently) draws a valuation from [0, 1]
What is a Bayes-Nash equilibrium for this?
Say a bidder with value vi bids vi(n-1)/n
Claim: this is an equilibrium!
Proof of equilibrium: suppose all others use strategy
For a bid b < (n-1)/n, the probability of winning is when the other n-1 bidders bid less than b. For a bidder to bid b, his valuation would be vi = bn/(n-1) Since he actually bids vi(n-1)/n = bn/(n-1)(n-1)/n = b
Because valuations are uniformly distributed (and the range is 0-1) the probability of anyone bidding b or less, is (bn/(n-1))
And the probability of all n-1 others bidding b or less is (bn/(n-1))n-1

21. First-price sealed-bid auction BNE
For a bid b < (n-1)/n, the probability of winning is (bn/(n-1))n-1, so the expected value is (vi-b)(bn/(n-1))n-1
Derivative w.r.t. b is (using product rule)
- (bn/(n-1))n-1 + (vi-b)(n-1)bn-2(n/(n-1))n-1 which should equal zero
- (bn/(n-1))n-1 + (vi-b)(n-1)bn-2(n/(n-1))n-1 = 0
Dividing both sizes by bn-2(n/(n-1))n-1
Implies -b + (vi-b)(n-1) = 0, which solves to b = vi(n-1)/n

22. Analyzing the expected revenue of the first-price auction
First-price auction: probability of there not being a bid higher than b is (bn/(n-1))n (for b < (n-1)/n)
This is the cumulative density function of the highest bid
Probability density function is the derivative, that is, it is nbn-1(n/(n-1))n
Expected value of highest bid is
n(n/(n-1))n∫(n-1)/nbndb = (n-1)/(n+1)
Upper bound of integral chosen as no bid will be higher

23. Revenue equivalence theorem
Suppose valuations for the single item are drawn i.i.d. from a continuous distribution over [L, H] (with no “gaps”), and agents are risk-neutral
Then, any two auction mechanisms that
in equilibrium always allocate the item to the bidder with the highest valuation, and
give an agent with valuation L an expected utility of 0,
will lead to the same expected revenue for the auctioneer

24. What if bidders are not risk-neutral?
Behavior in second-price/English/Japanese does not change, but behavior in first-price/Dutch does
Risk averse: first price/Dutch will get higher expected revenue than second price/Japanese/English
Risk seeking: second price/Japanese/English will get higher expected revenue than first price/Dutch

25. 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?

26. 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

27. 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?

28. 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.

29. 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.

30. 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'.

31. Redistribute as much of the VCG revenue as possible back to the agents
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)

32. Vickrey auction without a seller
) = 2 v(
) = 4 v(
) = 3
pays 3
(money wasted!)‏
But if there is no cost, people won’t be honest about their valuation

33. Can we redistribute the payment?
Idea: give everyone 1/n of the payment v(
) = 2 v(
) = 4 v(
) = 3
receives 1
receives 1
pays 3
receives 1
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

34. Incentive compatible redistribution [Bailey 97, Porter et al
Incentive compatible redistribution [Bailey 97, Porter et al. 04, Cavallo 06]
Idea: give everyone 1/n of second-highest other bids v(
) = 2 v(
) = 4 v(
) = 3
receives 1
Who pays what?
Strategy-proof
Your redistribution does not depend on your bid;
incentives are the same as in Vickrey

35. Incentive compatible redistribution [Bailey 97, Porter et al
Incentive compatible redistribution [Bailey 97, Porter et al. 04, Cavallo 06]
Idea: give everyone 1/n of second-highest other bids v(
) = 2 v(
) = 4 v(
) = 3
receives 1
pays 3
receives 2/3
receives 2/3
2/3 wasted (22%)‏
Strategy-proof
Your redistribution does not depend on your bid;
incentives are the same as in Vickrey

36. 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/n
Rn = V2/n

37. 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
Key point is their return is never a function of their bid
Idea pursued further in Guo & Conitzer 07 / Moulin 07