bayes nash implementation
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Bayes Nash Implementation
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Complete information games (you know the type of every other agent, type = payoff)◦ Nash equilibria: each players strategy is best response to the
other players strategies Incomplete information game (you don’t
know the type of the other agents)◦ Game G, common prior F, a strategy profile actions – how to play game (what to bid, how to answer…) ◦ Bayes Nash equilibrium for a game G and common prior F is a
strategy profile s such that for all i and is a best response when other agents play where
Complete vs. Incomplete
Bayes Nash Implementation There is a distribution Di on the types Ti of
Player i It is known to everyone The actual type of agent i, ti 2DiTi is the
private information i knows A profile of strategis si is a Bayes Nash
Equilibrium if for i all ti and all t’i Ed-i[ui(ti, si(ti), s-i(t-i) )] ¸ Ed-i[ui(t’i, s-i(t-i)) ]
Bayes Nash: First Price Auction
First price auction for a single item with two players.
Private values (types) t1 and t2 in T1=T2=[0,1] Does not make sense to bid true value –
utility 0. There are distributions D1 and D2 Looking for s1(t1) and s2(t2) that are best
replies to each other Suppose both D1 and D2 are uniform.Claim: The strategies s1(t1) = ti/2 are in Bayes
Nash Equilibriumt1
Cannot winWin half the time
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If ◦ Other agent bids half her value (uniform [0,1])◦ I bid b and my value is v
No point in bidding over max(1/2,v) The probability of my winning is 2b My Utility is he derivative is set to zero to
get This means that maximizes my utility
First Price, 2 agents, Uniform [0,1]
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Bayes Nash equilibria (assumes priors)◦ Today: characterization
Special case: Dominant strategy equilibria (VCG), problem: over “full domain” with 3 options in range (Arrow? GS? New: Roberts) – only affine maximizers (generalization of VCG) possible.
Implementation in undominated strategies: Not Bayes Nash, not dominant strategy, but assumes that agents are not totally stupid
Solution concepts for mechanisms and auctions (speical case of mechanisms) (?)
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Characterization of Equilibria
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What happens to the Bayes Nash equilibria characterization when one deals with arbitrary service conditions: ◦ A set is the set of allowable characteristic vectors◦ The auction can choose to service any subset of
bidders for whom there exists a characteristic vector
Prove the characterization of dominant truthful equilibria.
Homework #1
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Characterization of EquilibriaClaim1: If (¯1;¯2; : : : ;¯n) is a Bayes-Nash equilibrium, (agent i bids ¯ i (vi )
when vi is her value), then, for all i:1. The probability of allocation ai (vi ) is monotone increasing in vi .2. The expected utility ui (vi ) (expected utility of agent i when agents with
value vj bid bj (vj )) is a convex function of vi ,
ui (vi ) =Z vi
0ai (z)dz:
3. The expected payment
pi (vi ) = vi ai (vi ) ¡Z vi
0ai (z)dz =
Z vi
0za0
i (z)dz:
Claim2: If (¯1;¯2; : : : ;¯n) are such that either (1) and (2) hold or (1) and (3)hold then (¯1;¯2; : : :;¯n) are a Bayes-Nash equilibria.
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Let ui (w;v) bethe(expected) utility of agent i when shebids ¯ i (w) and hervalue is v. Let vi be the truevalue of agent i.
Choose some i, ¯x all ¯ j , j 6= i, u = ui , v = vi , a = ai .If ¯1; : : :;¯n is a Bayes Nash Equilibrium then
u(v;v) = va(v) ¡ p(v) ¸ va(w) ¡ p(w) = u(w;v):
But, if the true value of agent i was w we also get that
u(w;w) = wa(w) ¡ p(w) ¸ wa(v) ¡ p(v) = u(v;w):
Adding these two(v ¡ w) (a(v) ¡ a(w)) ¸ 0:
If v ¸ w then a(v) ¸ a(w). I.e., ai is monotonic for all i.
Claim 1 proof: Monotonic
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Claim 1 proof: Convex
u(v) = u(v;v) = supw
u(w;v) = supw
f va(w) ¡ p(w)g:
The supremum of a family of convex functions is convex
f convex:f (®x + (1¡ ®)z) · ®f (x) + (1¡ ®)f (z):
Ergo, is convexui (v)
If f : [a;b] 7! < is convex then it is the integral of it's (right) derivative
f (t) = f (a) +Z t
af +(x)dx:
where f+(x) is the right derivative at x
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Claim 1 proof: u’(v)=a(v), u(v) = int(a(z), z=0..v)For every v and w:
u(v) ¸ u(w;v) ¸ va(w)¡ p(w) = (wa(w) ¡ p(w))+(v¡ w)a(w) = u(w)+(v¡ w)a(w)
Or,u(v) ¡ u(w)
v ¡ w ¸ a(w):
If v approaches w from above, the left derivative u0(w) ¸ a(w). If v ap-proaches w from below the right derivative u0(w) · a(w). If u is di®erentiableat w then
u0(w) = a(w):
Since a convex function is the integral of it's right derivativewe have that
u(v) ¡ u(0) =Z v
0a(z)dz:
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Since
Claim 1, end
u(v) = va(v) ¡ p(v)p(v) = va(v) ¡ u(v)
pi (vi ) = vi ai (vi ) ¡Z vi
0ai (z)dz =
Z vi
0za0
i (z)dz:
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From (condition 3)
pi (vi ) = vi ai (vi ) ¡Z vi
0ai (z)dz =
Z vi
0za0
i (z)dz:
it follows that
u(v) =Z v
0a(z)dz:
u(w;v) = va(w) ¡ p(w) = (v ¡ w)a(w) +Z w
0a(z)dz:
As ai is monotonic (condition 1) this implies that
u(v) ¸ u(w;v):
Characterization: Claim 2 proof
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If bidding truthfully ( for all i) is a Bayes Nash equilibrium for auction A then A is said to be Bayes Nash incentive compatible
Bayes Nash Incentive Compatible Auctions
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For every auction A with a Bayes Nash equilibrium, there is another “equivalent” auction A’ which is Bayes-Nash incentive compatible (in which bidding truthfully is a Bayes-Nash equilibrium )
A’ simply simulates A with inputs ◦ A’ for first price auctions when all agents are
U[0,1] runs a first price auction with inputs The Big? Lie: not all “auctions” have a
single input.
The Revelation Principle
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Dominant strategy truthful: Bidding truthfully maximizes utility irrespective of what other bids are. Special case of Bayes Nash incentive compatible.
Dominant strategy truthful equilibria
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The probability of ) is weakly increasing in - must hold for any distribution including the distribution that gives all mass on
The expected payment of bidder i is
Dominant strategy truthful auctions
pi (vi ) = vi ai (vi ) ¡Z vi
0ai (z)dz =
Z vi
0za0
i (z)dz:
over internal randomization
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The probability of ) is weakly increasing in - must take values 0,1 only
The expected payment of bidder i is ◦ There is a threshold value such that the item is
allocated to bidder i if but not if ◦ If i gets item then payment is
Deterministic dominant truthful auctions
Expected RevenuesExpected Revenue:
◦ For first price auction: max(T1/2, T2/2) where T1 and T2 uniform in [0,1]
◦ For second price auction min(T1, T2) ◦ Which is better? ◦ Both are 1/3.◦ Coincidence?
Theorem [Revenue Equivalence]: under very general conditions, every two Bayesian Nash implementations of the same social choice function if for some player and some type they have the same
expected payment then◦ All types have the same expected payment to the player◦ If all player have the same expected payment: the expected
revenues are the same
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If A and A’ are two auctions with the same allocation rule in Bayes Nash equilibrium then for all bidders i and values we have that
Revenue Equivalence
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F strictly increasing If is a symmetric Bayes-Nash equilibrium
and strictly increasing in [0,h] then
◦ | This is the revenue from the 2nd price auction
IID distributions highest bidder wins
pi (vi ) = vi ai (vi ) ¡Z vi
0ai (z)dz =
Z vi
0za0
i (z)dz:
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w
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First price auctions
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n bidders U[0,1]