Game theory review
• A game is a collection of players, the actions those players can take, and
their preferences over the selection of actions taken by all the players
• A strategy s∗i is dominant for player i if, given any selection of strategies
a−i by i’s opponents, s∗i maximizes i’s payoff
• If all the players are playing dominant strategies, they are playing a purestrategy Nash equilibrium of the game
• We’d like to design games for which the players’ dominant strategies are
simple, and the outcomes are desirable. Today we’ll look at the quintessential example of such a game.
What are auctions?
They are games.
• Players: the bidders, denoted i = 1, 2, ..., N
• Actions: their bids, b1 , b2 , ..., bN
• Payoffs: if buyer i wins, he gets a payoff vi − ti , where vi is bidder i’s value
and ti (b1 , b2 , ..., bN ) is a payment (not necessarily his bid), and if buyer i
loses, he gets a payoff of zero
The English Auction
This is the most important game in all of economics.
• The price clock starts at zero
• The auctioneer raises the price clock slowly, allowing agents to indicate
whether they want to continue or withdraw
• When the second-to-last buyer withdraws from the auction, the winner’s
payment is set equal to the current price on the clock (call it b(2) ), and
the winner is the buyer who failed to withdraw
The English Auction
How should bidders behave in the English auction?
The English Auction
Theorem 1. It is a dominant strategy to drop out of the English auction at
bi = v i .
If a buyer bids bi = vi , we say he bids honestly.
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The English Auction
The key to the auction is that you control the likelihood you win, but not
the price you pay, which will always be the lowest price you could pay and still
be a winner (meditate on this).
• What if i was to stay in until bi > vi ? There’s two cases: (1) i would have
won when bidding bi = vi , and (2) i would have lost when bidding bi = vi .
– Case 1: if i would have won when bidding bi = vi , then the nexthighest bid is less than vi , so i makes the same payment when bidding
bi > vi and i’s payoff doesn’t change.
– Case 2: if i would have lost when bidding bi = vi , then the nexthighest bid is greater than vi , so i makes a loss now that i wins with
a bid above vi .
In either case, i is weakly better off bidding honestly.
The English Auction
• What if i was to bid bi < vi ? There’s two cases: (1) i would have won
when bidding bi = vi , and (2) i would have lost when bidding bi = vi .
– Case 1: if i would have won when bidding bi = vi , then the nexthighest bid is less than bi , so i makes the same payment when bidding
bi = vi and i’s payoff doesn’t change.
– Case 2: if i would have lost when bidding bi < vi , then the nexthighest bid is greater than bi but less than vi , so i could have gotten
a strictly positive payoff instead of zero.
In either case, i is weakly better off bidding honestly.
• If bi > vi and bi < vi are both dominated by bidding bi = vi , then bi = vi
is a dominant strategy.
The English Auction
Why is the English auction so important?
• Dominant strategies: the buyers have a dominant strategy to bid honestly
• Efficiency: the buyer with the highest value wins
• Stability: if the price paid by the winner were any lower, some other buyer
could rightfully object
• Individual rationality: since bi ≥ b(2) , any winner’s payoff is vi − b(2) ≥
vi − bi ≥ 0, so no buyer will regret winning
• Privacy preserving: if the auction ends then the buyer with the secondhighest value drops out, no one ever learns the winner’s true value
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• Robust: players’ optimal strategies do not depend on their beliefs about
their opponents
It is easy to participate, and the outcome satisfies many desirable properties.
The Second-price Auction
• The English auction is an “open” format: the buyers indicate their interest
in participating over time
• Sometimes, this is infeasible or undesirable, and a “closed” or “silent”
format is adopted
• It retains most of the positive features of the English auction, except,
potentially, for privacy preservation of the winner’s value
The Second-price Auction
In the second price auction (SPA),
• Each buyer i submits a bid bi
• The highest bidder wins, but pays the second-highest bid
A buyer bids honestly if bi = vi .
Theorem 2. It is a dominant strategy to bid honestly in the second-price auction.
The Second-price Auction
• Charity auctions, for example, often use “silent auctions” to sell donations
for cash, which are essentially second-price auctions
• Many Internet-based sales platforms use second-pricing, or a generalization of it (Google AdWords)
• Running an English auction has high transactions costs (Coase)
What is a second-price auction?
Let pi (bi , b−i ) be the probability that i wins, given that he bid bi against
opponents bidding b−i . In the SPA, i’s payment is constructed as
ti (bi , b−i ) = −
X
pj (bi , b−i )bj
j6=i
|
{z
}
Welfare of the other agents if i participates; zero if i wins
+
X
pj (bi = ∅, b−i )bj
j6=i
|
{z
}
Welfare of the other agents if i were to opt out; second-highest bid/value if i wins
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If i wins, the welfare of the other agents if i participates is zero, while the welfare
of the other agents if i were to opt out is the second highest value when i wins;
if i loses, i has no impact on welfare in either case, and makes a payment of
zero. This is a generalization of Lindahl pricing; we call it Vickrey pricing, for
the economist who figured out how to incorporate private information into the
problem of selling goods.
Other concerns
• Revenue maximization — is this the “best” way to sell a good if I want
to maximize profits or minimize costs?
• Multiple units — what if the seller has multiple units and the buyers still
only want one? What if the seller has multiple units and the buyers all
would like multiple units? What if the seller has multiple heterogeneous
goods and the buyers have different valuations over each?
Let’s look at revenue maximization first.
Revenue maximization
What’s the worst thing that can happen to a seller in the SPA?
• Some buyer i submits a yuge bid, but...
• ...no one else does, so the good is sold for a very low price despite the
buyer having a really high value for it.
• Basicaly, the seller inadvertently faces a monopsonist, and would be committed to trading at a price of zero.
To protect sellers from bad outcomes like this, we add reserve prices.
Revenue maximization
In the second price auction with a reserve price (SPAR),
• The seller sets a reserve price r
• Each buyer i submits a bid bi
• The highest bid greater than the reserve price wins. The winner pays the
maximum of the second-highest bid and the reserve price.
A buyer bids honestly if bi = vi .
Theorem 3. It is a dominant strategy to bid honestly in the SPAR.
But is this the best way to sell a good? Can it raise the most revenue?
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The First-price Auction
To appreciate the profit maximization question, instead of the second-price
auction, consider the first-price auction with reserve (FPAR):
• The seller sets a reserve price r
• Each buyer i submits a bid bi
• The highest bid greater than the reserve price wins. The winner pays his
bid.
Does this raise more or less revenue than the second-price auction?
The First-price Auction
• In the FPA, buyers solve
max p(bi )(vi − bi )
bi
where p(bi ) is the probability of winning, given a bid of bi
• Is honest bidding a good strategy?
• The FONC is
∆p(bi )(vi − bi )
{z
}
|
Benefit of a higher likelihood of winning and getting a payoff
−
p(bi )∆bi
| {z }
= 0,
Cost of paying a higher bid, conditional on winning
just like a monopolist. (We’ll skip over how to solve this exactly for the
moment, trust me that I know how to do it)
• In the SPA, agents bid honestly (+) but only pay the second-highest bid
(-). In the FPA, agents shade their bids down (-) but pay the highest bid
(+). Which of these effects dominates?
Simulations
• https://marketdesign.shinyapps.io/simulation1/
• https://marketdesign.shinyapps.io/simulation2/
What happens to revenue as the number of bidders increases, in particular?
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Revenue equivalence
Theorem 4. Any method of selling a single good to buyers with independent,
private values that gives the good to the agent with the highest value above the
reserve price and charges the losers nothing raises the same amount of revenue.
So the FPAR and SPAR with the optimal reserve price are both optimal in
the class of all possible ways of selling things.
Procurement auctions
As a tool of public policy, auctions are incredibly popular. Suppose the
government is trying to procure some good (like a bridge or computer or fighter
plane) that it values at v. It wants to buy at the lowest price it can. The reverse
auction or procurement auction is the game where
• i = 1, 2, ..., N sellers each submit a bid bi
• The lowest bidder wins, and is paid the second-highest bid
We can also impose a reserve price: if all the losing bids are above r, the winner
is paid only r. The efficient reserve price would be that any winning bid must
be less than v, so the government doesn’t pay more than the good is worth.
Collusion in auctions
• This assumes that the players in the game are not colluding or otherwise
working together
• How should firms collude to maximize their payoffs in a standard SPAR?
In the reverse auction?
• Porter and Zona (1992) considered collusion at procurement auctions in
Long Island for highway construction projects, and look for this “strategic
over-bidding”
• ”We all sat at the conference table one of the contractors would have a list of
upcoming contracts ... they’d talk about the contract ... how much money who
won the last one ... who should get this one ... The contractors who were tagged
to be the low bidders would work out their ”We all sat at the conference table
one of the contractors would have a list of upcoming contracts ... they’d talk
about the contract ... how much money who won the last one ... who should
get this one ... The contractors who were tagged to be the low bidders would
work out their bid figures ... The rest of the contractors would then come up
with higher bids.”
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Collusion in auctions
• Porter and Zona look at the backlog of work the firm faces, and how
intensely it is currenlty utilizing its capactiy: these firms should have
higher marginal costs of an additional job, and should bid higher
• They find that firms bidding lower have a stronger correlation between
cost shifters and their bid, while the bids made by firms who big higher
have little connection with anything
• “Unfortunately, if an antitrust authority or procurement agency were to
publicly announce the adoption our test procedure, it would be relatively
easy for an effective cartel to tailor its phantom bids to disguise collusive
behavior. For example, all cartel firms could scale their competitive bids
up by the same percentage. The bid ranking would then coincide with
cost rankings.”
Other issues
• Risk aversion: favors the FPAR
• Learning from others’ bids: favors the SPAR
• Budget constraints on the bidders: favors the SPAR
• Regret aversion: favors the SPAR
Summary from last time
• Auctions are important market designs: buyers or sellers submit bids,
highest or lowest bids win, and winners make a payment that potentially
depends on their bid or those of others
• The second-price auction with reserve (SPAR) is the game where (1) the
seller sets a reserve price, (2) each buyer submits a bid, (3) the highest
bidder above the reserve wins, but pays the maximum of the reserve price
and the second-highest bid. The winner gets a payoff of vi − max{r, b(2) },
and the losers get payoffs of zero
• It is a weakly dominant strategy to bid honestly in the SPAR, the outcome is efficient if r = 0, and profit-maximizing if the reserve price is set
optimally
• The SPAR can easily be turned into an auction to buy rather than to sell:
the lowest bidder wins, and receives the minimum of the second-lowest bid
or the reserve price. This is great for governments who want to procure
goods like construction projects.
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Auction-like markets
• eBay looks like a second price auction: http://www.ebay.com/
• The seller can set a secret reserve price, buyers can make open bids or use
a proxy bid (a robot that bids for them up to a certain limit), and the
auction ends at a pre-specified time
• Bid-sniping: enter bids right at the end of the auction in order to “steal”
the good from the current standing high bidder
• “40 percent of all eBay-Computers auctions and 59 percent of all eBayAntiques auctions as compared to about 3 percent of both Amazon-Computers
and Amazon-Antiques auctions, respectively, have last bids in the last 5
minutes. The pattern repeats in the last minute and even in the last ten
seconds. In the 240 eBay-auctions, 89 have bids in the last minute and
29 in the last ten seconds. In Amazon, on the other hand, only one bid
arrived in the last minute.” (Roth and Ockenfels, 2002)
Auction-like markets
• In a “penny auction” or “all-pay auction”, we start at a price of zero. Each
bidder pays a bid increment to “stay in”. Once all a bidder’s competitors
have dropped out, he is declared the winner and given the good
• This is a popular eCommerce business model
• It is totally evil. It is not an “auction”, it is a “war of attrition”.
Auction-like markets
• The Better Business Bureau warns consumers, “although not all penny
auction sites are scams, some are being investigated as online gambling.
BBB recommends you ... know exactly how the bidding works, set a limit
for yourself, and be prepared to walk away before you go over that limit.”
• The idea is that the goods might sell for low prices: $30 for a tv, say.
But hundreds or thousands of people each bid a few bucks on it, so the
company is making a ton of money in the background. These losses add
up for the poor people involved. It’s really a lottery.
Multiple unit auctions
There are a lot of important multiple unit auctions, where the market designer is looking to sell or buy more than one unit at a time:
• https://www.iso-ne.com/
• http://wireless.fcc.gov/auctions/default.htm?job=auctionsh omehttp : //www.ppaghana.org/
But in the SPAR, only one unit is sold or purchased at a time. We want to sell
or buy more than one unit at a time.
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Clean Power Plan
Here’s a case study that’s in court right now: https://www.epa.gov/cleanpowerplan/cleanpower-plan-existing-power-plants
• Using the Clean Air Act, the Federal government has set emissions targets
for power generation for each state in the country that need to be met by
2022, 2029, and 2030
• The reality of the law is that it requires each state to close a certain
number of coal-burning power plants
• The Federal Government has advocated using cap-and-trade or other marketbased mechanisms
• Many actors have sued the Federal Government over this; regardless of
how those lawsuits are decided, many actors will then sue the states when
they try to implement their plans
Clean Power Plan
• The expected discounted profit of a plant to owner i is πi , i = 1, 2, ..., N ;
suppose they are all equally “dirty”, to keep the problem simple, but they
have varying levels of investment that make some more profitable/efficient
than others
• The market is individually rational only if the owner receives at least πi
for agreeing to close his plant down; otherwise there’s a lawsuit
• The government has told us we need to close down at least K plants,
K<N
• We want to find the cheapest way to induce the least profitable plants to
exit the market, such that we satisfy the Federal Government’s emissions
constraint
Multiple units
• The second price auction gives us an important principle: if you control
the likelihood you win, but not the price you pay, it’s possible to give you
a dominant strategy to bid honestly
• We can only implement the efficient outcome (closing down the least profitable plants) if we know the agents’ true valuations, which are only known
to the agents themselves
• But we can extend the basic idea of the SPAR pretty easily here: all of
the winners pay (receive) the bid of one of the losers
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Highest-Rejected Bid (HRB) Auctions
Suppose we want to sell K units to N buyers, N > K:
• All the buyers i = 1, 2, ..., N submit a bid, bi . Order the bids from highest
to lowest, b(1) ≥ b(2) ≥ ... ≥ b(N ) .
• The, at most, K buyers with bids above a reserve price r win, but they all
pay the maximum of the highest losing bid or the reserve price. A winner
then gets a payoff of vi − max{b(K+1) , r}, where vi is i’s value of winning
the good.
Notice, it is a dominant strategy to bid honestly in the HRB auction, just like
the SPAR.
Lowest-Rejected Bid (LRB) Auctions
Suppose we want to buy K units from N sellers, N > K:
• All the sellers i = 1, 2, ..., N submit a bid, bi . Order the bids from lowest
to highest, b(1) ≤ b(2) ≤ ... ≤ b(N ) .
• The, at most, K sellers with bids below a reserve price r win, but they
all receive the minimum of the lowest losing bid or the reserve price. A
winner then gets a payoff of min{b(K+1) , r} − ci , where ci is i’s cost of
providing the good.
Again, it is a dominant strategy to bid honestly in the LRB auction.
The Clean Power Plan
• How do we adapt the LRB to the CPP problem?
• So we can induce the least profitable/most inefficient plants to leave the
market, and we implement the efficient outcome in dominant strategies.
• What is the new political problem that emerges? Thoughts? Strategies
(how could I drive down power plants’ values of their firms without throwing the money away)?
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