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Algorithmic Applications of Game Theory

Lecture 8

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Reading

• Chapter 9 and chapter 11 in Algorithmic Game Theory.

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Reminder - model

• Each bidder i has a valuation function vi which takes a subset of the objects, and says how much the bidder wants this subset– In the simple case of a single item vi is just how much the bidder

values the item

• Bidder i will pay a price pi. The utility (happiness) of bidder i is denoted ui, withui(xi) = vi(xi) - pi

where bidder i received xi (either the item or an empty set)• Auctioneer does not know the bidder’s valuation function!

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Second price auction

• Winner is highest bidder• Price is the second highest bid

• This is what you really get in the ascending auction

• Dominant strategy truthful

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Auctioning more than one good

• A player i has an additive valuation function if for every two sets of good S, T

• If all players have additive valuations, independent auctions make sense.

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Non additive valuations

• I want to have a salad with tomatoes and cucumbers. I don’t want just tomatoes or just cucumbers. So:v({cucumber, tomato}) > v({cucumber}) + v({tomato})

• I want to have a fruit – either a banana or an apple, and I don’t care which. So:v({A,B}) = v({A}) = v({B}) < v({A}) + v({B})

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Dominant strategies

• Suppose we have an auction, in which each player has a dominant strategy for every valuation function vi he has

• Revelation Principle: there exists a mechanism in which it is dominant for every player to report his valuation function vi

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Efficiency

• Usually easier than revenue• Defined as the sum of utilities of the players and

the utility of the auctioneer

• But the utility of player i is vi(xi) – pi

• And the utility of the auctioneer is pi

• So the sum of utilities is vi(xi) which is independent of the prices

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Why do we need prices then?

• Each player will report the value of each bundle to her

• The auctioneer will compute the best allocation, and this will be the outcome

• Problem: players will have incentive to report that the goods are very valuable to them

• We need prices to elicit the true values from the players

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VCG auction

• Suppose we can find the optimal solution if we had all the inputs from the players

• Lemma 1: If each player i gets paid

then the mechanism is truthful• Lemma 2: The mechanism remains truthful

even if player i pays an additional hi(v-i).where hi(v-i) is any function which is does not depend on the bid of player i.

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Making money

• The easiest way to think of VCG is set hi(v-i) = 0.

• But then the auctioneer has to pay each player a lot of money

• We want the payments to be such that– Players don’t loose from the auction– The auctioneer does not pay money to any player

• Use

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Clarke’s payment

• The rule

satisfies:– No player looses from participating in the auction– The auctioneer never pays a player

• Meaning: player i is charged the damage that he caused to the world

h 𝑖 (𝑣−𝑖 )=∑𝑗 ≠𝑖

𝑣 𝑗(𝑂𝑃𝑇 (𝑣−𝑖 ))

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Examples of VCG – Public Project

• VCG is not just for auctions… The government can build a bridge, which would cost C

• Each citizen has value vi for the bridge

• We want to build the bridge iff vi ≥ C• Use VCG to decide. Plugging in Clarke’s payment rule

gives that i pays max(0, C-ki vk )

• Each player pays at most vi

• The government will get its money back iff vi = C

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Example of VCG - trade

• A seller has an object, values it for vs

• Buyer values it for vb

• We want them to trade if vb > vs

• Doing this by VCG, means that the mechanism charges vs from the buyer, and pays vb to the seller, loosing money…

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We are not making much money here…

• Bummer• If you want citizens not to pay if the bridge is

not built, or buyers not to pay when there is no trade, then these are the only possible prices

• Getting revenue truthfully is hard (and interesting!)

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Disadvantages of VCG

• Complexity – the agents need to describe their valuation for every possible bundle (m items 2m numbers)

• Even if valuations have succinct representations, the auctioneer has to find the optimal solution for VCG to be truthful– If you have an approximation algorithm, you loose

truthfulness

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Is it hard to find the optimal solution?

• Yes.• A bidder i is single minded if there exists a set S i

such that vi(S) = ai if SiS, and vi(S)=0 otherwise• Theorem: Even if all the bidders are single

minded, it is NP hard to find the optimal solution.

• Moreover, even finding an approximate solution which would generate more than OPT/m0.5 welfare is NP hard

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Proof

• Reduction from independent set• Given a graph G with m edges and n vertices:

– Each edge becomes an item– For each vertex we add a player who wants exactly the set of

items adjacent to each, and has value 1 for that set• If two vertices are adjacent, the corresponding players can

not be satisfied together– Getting a welfare of k gives an independent set of size k

• But approximating independent set better than n1- is NP hard– So approximating better than m1/2 - is also NP hard