Computational Issues in Game Theory Lecture 2: Auctions

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Computational Issues in
Game Theory Lecture 2
Auctions

• Edith Elkind
• Intelligence, Agents, Multimedia group (IAM)
• School of Electronics and CS
• U. of Southampton

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Plan

• Single-item auctions
• Multi-unit auctions
• Mechanism design
• Combinatorial auctions
• Combinatorial procurement auctions

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Single-item auctions
classic auctions

• English auction
• bidding starts at 0, bidders submit bids in turn
• new bid has to exceed the current bid
• the last bidder wins, pays his bid
• Dutch auction
• bidding starts high,
  auctioneer lowers the price
• the first bidder to accept the price wins

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Single-item auctions
sealed-bid auctions

• Sealed-bid auctions all bidders submit their
  bids simulateneously in envelopes
• The bidder who submits the highest bid wins and
• pays his bid
  (first-price auction)
• pays second-highest bid
  (second-price auction)

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English auction vs. second-price auction
strategic equivalence

• English auction bidding stops when the 2nd
  highest bidder drops out
• Highest bidder wins, pays (almost) 2nd highest
  bid

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Dutch auction vs. first-price auction strategic
equivalence

• First-price auction have to decide how much to
  pay in absence of any information about other
  bidders
• Dutch auction have to decide when to accept in
  absense of any information about other bidders
• as soon as you learn anything about others, the
  auction is over

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Standard assumptions

• IPV independent private values
• all bidders draw their values from
  the same distribution
• independently at random
• Solution concept Bayes-Nash equilibruim
• cannot improve expected profit by deviating
• Practice
• bidders can be asymmetric
• dependent values

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What is a good auction?

• Allocative efficiency
  in all these auctions,
  highest bidder always wins (no reserve prices)
• Computational efficiency
  open-cry takes longer than sealed-bid
  
• Revenue should we choose 1st price
  or 2nd price auction?
• or something else entirely?

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Revenue Equivalence Theorem

• Any two auctions such that
• the bidder with the highest value wins
• bidder with the lowest value expects 0 profit
• bidders are risk-neutral
• value distributions are strictly increasing and
  atomless
• have the same revenue!
• also same expected profit for each bidder
• all 4 auction formats we considered - and more!
• which auctions are not revenue-equivalent to
  English auction? reserve prices, entry fees

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Bidding strategies

• English auction
  dominant strategy to bid truthfully
• 2nd price auction
  dominant strategy to bid truthfully
• proof suppose your value is v,
  the highest value among others
  is v
• winners payment is the lowest amount
  he can bid and still win

v
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First-price auction how to bid?

• Bidding truthfully is not optimal
• Optimal strategy depends on the number of other
  players
• if you are the only player, bid e
• if there are many players, shade very little
• Claim optimal bidding strategy is given by
  b(x) EY1 Y1 lt x,
  where Y1 max of other players values

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First-price auction how to bid?

• b(x) EY1 Y1 lt x
• Proof
• let G be the distribution of Y1 maxj?ivj
• Expected profit from bidding b when others use
  b(x) G(b-1(b))(x - b)
• write down FOC wrt b, replace b with b(x)...
• Alternative proof RET
• in 2nd price auction, bidders expected payment
  is EY1 Y1 lt x

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Maximizing revenue Example 1

• 2 bidders A and B
• As value 4 w.p. 50, 5 w.p. 50
• Bs value 10 w.p. 90, 6 w.p. 10
• 2nd price auction 5 w.p. 50, 4 w.p. 50,
• expected profit 4.5
• if B bids 10, he wins and pays 10,
  else A wins and pays 4
• truthful
• expected profit 9.4

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Maximizing revenue Example 2

• 2 bidders, uniform distributions on 0, 1
• 2nd price auction revenue 1/3
• expected 2nd highest bid 1/3
• 2nd price auction with reserve price of 1/2
• w.p. 3/4, at least one bidder has value gt1/2
• expected profit at least 3/8 gt 1/3
• Is 1/2 the optimal reserve price?
• More generally, what is the
  optimal auction format?

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Optimal auctions M81

• xi Di with PDF fi and CDF Fi
• Virtual valuation ci(xi) xi
• if nonmonotone,
• flatten out
• Allocation rule
• to the bidder with the highest nonnegative
  virtual valuation
• Payment rule
• the lowest amount one can bid and still win
  (threshold payment)
• Example all values are from U0, 1
• c(x) 2x -1 gt allocate to the highest bidder
  whose
  value is above 0.5

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Multi-unit auctions

• m3 identical items, 5 unit-demand bidders
• values 1, 5, 6, 8, 11
• (m1)st price auction
• sell to top 3 bidders, charge 5 - truthful!
• m3 identical items, a non-unit-demand bidder
• bidder X with demand 2 and values 5 and 11
• 3 bidders with values 1, 6, 8
• (m1)st price auction no longer truthful
  X prefers to bid
  (1, 11)

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Multi-unit auction VCG mechanism

• Idea the price paid by i should be
  the negative externality i
  imposes on others
• 2nd price auction if winner were not present,
  2nd highest bidder
  would get the object
• alternative social welfare 2nd highest value
• X 5, 11 Y 1, Z 6, T 8
• with X SW of others 6814
• without X SW of others 16815
• with T 17, without T 22, so T pays 5
• with Z 19, without Z 24, so Z pays 5

X pays 1
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Mechanism Design

• Set of possible outcomes O O1, , Om
• n agents, each has a privately known type ti
• utilities u(Oi, tj)
• Space of allowable bids
• direct mechanism bids specify types
• mechanism (allocation rule, payment rule)
• allocation rule bids ? outcomes
• payment rule bids ? payments
• Goal implement a social choice function
• mapping from values to outcomes

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Examples

• Voting
• outcomes winners
• types preferences over candidates
• goal e.g., select a candidate that is listed in
  top 10 by the largest of voters
• Auctions
• outcomes allocations
• types valuations for bundles
• goal e.g., maximize social welfare or revenue

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Implementation

• mechanism M (A, P) is
  a dominant-strategy implementation
  of a social choice
  function f if
• for bidder i with type ti it is
  a dominant strategy to
  bid bi and
• A(b1, , bn) f(t1, , tn)
• mechanism is truthful if bi ti
• implementation in NE strategies similar

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Revelation Principle

• Truthfulness is free given a mechanism
  M (A, P) that has dominant strategies,
  we can design a truthful mechanism M that for
  every bid vector chooses the same outcome and
  pays the same amounts to all parties.

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VCG general case

• i bids X, others bid Y1, , Yk.
• O1 optimal outcome for X, Y1, , Yk
• O2 optimal outcome for Y1, , Yk
• Allocation rule maximize social welfare (O1)
• Payment rule pi Sj?ivj(O2) -Sj?ivj(O1)
• SW of others in the opt outcome without i -
  SW of others in the opt
  outcome with i
• is utility
  vi(O1) -
  Sj?ivj(O2) -Sj?ivj(O1) SW(O1) - SW(O2)
• i wants to maximize SW(O1) gt bids truthfully
• Hence, VCG is truthful!

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VCG other applications

• VCG can be used whenever
• goal is to maximize social welfare
• payments are allowed
• Public project should we build a pool?
• 2 outcomes build, not build
• player i values build at vi, not build at 0
• cost C (or C/n per player)
• SW(build) Sj vj - C, SW(not build) 0

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VCG properties

• pi SW(O2) - (SW of others in O1)
• individually rational pi vi
• pi - vi SW(O2) - SW(O1) 0 (O1 is optimal)
• truthful
• NOT budget balanced
• public project example
• cost 3, 5 players with value 1 each
• total revenue 0

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Is there an alternative to VCG?

• VCG always maximizes social welfare
• Myerson optimal auction is not VCG
• VCG may not be budget-balanced
• VCG requires players to communicate valuations
  for all outcomes to the center
• may be computationally infeasible

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Combinatorial Auctions

• Several distinct items for sale
• Buyers have valuations for bundles of items
• not necessarily additive
• complements v(x, y) gt v(x)v(y)
• plane tickets and hotel rooms
• substitutes v(x, y) lt v(x)v(y)
• movie tickets and theater tickets

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Combinatorial auctions representation

• Each bidder has to specify his value for each
  bundle 2n numbers
• Compactly representable valuations
• single-minded bidders there is a bundle B such
  that v(X)v for any X s.t B ? X and 0 otherwise
• additive with budgets
• Can use VCG
• ? truthful each winner pays his threshold bid
• ? NP-hard, but practical algorithms exist
• ? revenue can be low
• Can design optimal auctions (virtual values)

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Combinatorial auctions non-direct revelation
mechanisms

• When bidders are not single-minded,
  non-VCG-like mechanisms might still
  work
• simultaneous auctions
• sequential auctions
• Examples
• gross substitutes if xs price goes up, demand
  for y does not go down
• submodular v(xUS) - v(S) v(xUT) - v(T) for
  S?T
• both may need exp(n) bits to describe, yet

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Gross substitutes valuations

• Idea run m simultaneous English auctions
• no bidder will ever walk away from
  an auction where he is current winner
• GS property
• Prices increase at each step,
  so eventually converge
• Truthful
• Finds optimal allocation

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Submodular valuations

• Idea auction off items one by one
  using 2nd price auctions
• myopically truthful
• 2-approximation to social welfare
• Proof
• (S1, , Sn) - output of our algorithm
• (T1, , Tn) - optimal allocation
• SW(S1UT1, , SnUTn) 2SW(S1, , Sn)
• allocate according to (S1, , Sn) first
• 2nd copy of each item is less useful than the 1st

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Combinatorial procurement auctions an example
4
2
0
1
2
S
T
12
10
2
5

• The goal find a path with the smallest cost.

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Formal model

• intuition need to hire a team of workers
• 1 buyer, n sellers E 1, , n
• Buyer wants to hire a feasible set (team) Si
  FS1, , Sm, Si ? E
• monopoly-free n Si Ø
• Sellers have costs
• ce if selected, 0 otherwise
• the cost ce is known to e only

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Examples

• Path auctions
• network G (V, E, s, t)
• agents are edges (E E)
• feasible sets are s-t paths in G
• Vertex cover auctions
• graph G (V, E)
• agents are vertices (E V)
• feasible sets are vertex covers for G
• List representation
• E a, b, c, d, e, f
• F a, b, a, c, d, e, f, b, e

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VCG for path auctions

• Allocation rule pick the cheapest path (in terms
  of bids).
• Payment rule a losing edge gets 0, a winning
  edge gets t, where t is the highest bid at which
  it still wins (threshold bid).

3
2
0
a can raise its bid to 6 and still win, so a
gets 6.
b
a
c
S
T
7
2
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VCG Good and Bad

• Good
• always choose the shortest path
• truthtelling is a dominant strategy
• Bad huge payments
• each edge on the upper path can raise its bid by
  d
• the total payment is Lnd.

Can we do better?
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Minimizing total payment

• Frugality ratio ratio of total payment and
  the cost of second cheapest
  path
• Our example frugality ratio of VCG n
• Theorem Elkind et al., 2004 frugality ratio of
  any dominant strategy mechanism is n/2
• VCG is not as bad as it seems
• Theorem Elkind et al., 2004
  can design an optimal
  mechanism (virtual costs)
• if we know cost distributions

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Costs of cheap labor Elkind 2005

• Can decrease VCG payments by deleting edges
• Reducing competition in the market leads to lower
  prices???
• How can we decide which edges to delete?
• NP-hard even for fairly simple graphs

VCG on G n/2(n-n/2)n/2 n2/4n/2
VCG on G\Q n0n n
Edge deletion reduces payments by a factor of
W(n)!
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Conclusions

• Single-item auctions are well understood
• Multi-item and combinatorial auctions less so
• Mechanism design powerful tools
• revelation principle
• VCG
• Computational considerations matter
• Active research area