Mechanism Design

Ruta Mehta

Game design (not video games!) to achieve a desired goal, like fairness, social welfare maximization, etc.

Widely Applicable

Lets Focus on Voting

• Setting– Candidates a, b and c– Voters with preference order over {a, b, c}

• Select the majority choice

•

Need for more complex mechanisms

-> Condorcet’s Paradox!

First Preference Majority

• Example:

– • Winner: a

Encourages Strategic Voting!

→𝑐¿4′ 𝑏¿4

′ 𝑎

Tie between a and c

Borda Count

• Give weight to -th candidate in a preference order.

• Take sum over all the voters. • The one with highest sum wins.

Is this strategy proof?NO!𝑎 ¿1𝑏¿1𝑐 ,𝑎¿2𝑐 ¿2𝑏 ,𝑏¿3𝑎¿3𝑐 ,𝑐¿4𝑏¿4𝑎

𝑏¿4𝑐¿4𝑎

In General: Social Choice

• Setting (Recall):– Set of choices. agents (voters).– Set of all complete preference orders over

• Social Choice Function:– Function

Desired Properties of f

• Incentive-compatible(IC) (strategy proof)

• Monotone– then and

IC iff Monotone• No dictators– No such that is always the first choice as per .

Gibbard-Satterthwaite

Theorem: If a social choice function is incentive-compatible, where |A|>=3, then it is dictatorship.

“Field of Mechanism Design attempts escaping from this impossibility result using various modifications in the model.”

Introducing Payment

Single Item Auction:• N bidders. • The item is worth to bidder .• Bid of bidder is .• Incentive Compatible:– If then is best.

• IC (in Dominant Strategy):– is the best bid regardless of what others

do.

Auction

• Item goes to the bidder with highest bid.• Payment:– No payments

Best to bid

– The winner pays his bid, rest nothingSuppose and then

– The winner pays the second highest bid, rest nothing (Vickery)

Not even IC

Vickery Auction is IC (in DS)

Proof on board

Open-Outcry Auctions

• English Auction: Start with a low price and keep increasing until only one buyer is interested.– Equivalent to Vickery auction!

• Dutch Auction: Start with a very high price where no one is interested. Keep decreasing until someone gets interested.– Equivalent to first price auction!

Extensions?

• What if identical items to be auctioned?• What if an item has to be procured?

In general how to extract private information from agent to make social optimal decision?

General Setting

• : Set of alternatives to choose from• : Set of valuation functions () for agent • Each agent reports (may not be true)• The mechanism– Social choice function: – Payments

Example: Single Item Auction

• ={1-wins, 2-wins, …, n-wins}

• True valuation:

What is the key idea in Vickery Auction? How to extend that to the general setting?

Vickery-Clarke-Groves Mechanism

• Let be reported valuations.• Social choice func: Social welfare maximizing

• Payments: Align payoff with SW maximization– (Independent of )

Properties of VCG

• Incentive Compatible (in DS)

• Individually rational? – Non-negative payoff:

• No positive transfers?

Proof on board

No if

No if

𝑝𝑖=h𝑖 (𝑣− 𝑖 )−∑𝑗 ≠𝑖

𝑣 𝑗(𝑎∗)

Clarke Pivot Rule

Pay the damage you cause• captures maximum social welfare excluding –

• -

• IR: YES!• No positive transfer: YES!

Others welfare without i

Others welfare with i

Example: Single Item Auction

• Recall:– ={1-wins, 2-wins, …, n-wins}

– True valuation: • Report: , zero elsewhere• Social choice: – Highest bid

• Payment:– Verify: Winner pays second highest bid

Example: Multiunit Auction

identical items, each agent wants one unit.• Agent values it at and reports • Alternatives? • Set of valuation functions and bid s?

– Report: • Winners?

– Bidders with highest bids win• Payments?

– Each winner pays highest bid

Example: Reverse Auction

Procure an item from agents• Agent reports cost Break:• Winner?• Payment?

Example: Resource Allocation

Buying an s-t path in a network• , each edge is owned by an agent• Edge costs to its agent (value )• Cost of a path is • Social welfare maximizing path: Shortest path • Payment to edge :– If then zero– Else, let be the shortest path in .

Example: Multi Item Auction

different items on auction. • Agent values set , nothing more nothing less

(Single minded bidder)– Value of set is if , else zero.

• Agent reports

• Computation of social welfare maximizing allocation is NP-hard!– Even approximation On board

Approximate Algorithm

• Reorder: . Let • For each – If then

• Payments: For – ; is smallest s.t. and

• Observe • is the lowest possible bid to win.

Incentive Compatible

• Suppose (S’,w’) gets better payoff than (S,w)– and (S’,w’) should win

• Then (S,w’) should be better– betters the chances of winning and may decrease

payment • If (S,w) winning bid then for (S,w’)– w’>w does not help; w’<w may loose

• If (S,w) loosing bid then for (S,w’)– w’<w does not help; w’>w may fetch –ve payoff

approximate social welfare

Break: Show that Recall:

Pros and Cons of VCG

• Best for bidders– Government auctions like road contract,

bandwidth allocation• May not be efficiently computable– Multi item auction

• Worst for auctioneer– May get zero payment!

Sponsored Search Auctions

Ad Auctions

• Generalized Second Price (GSP)– Google, Yahoo, Bing

• Bid on keywords– If the user query

contains your keyword, your bid qualifies for the auction

GSP Auction Setting

• bidders, slots• Agent values a click with and bids – Suppose

• is the probability that user clicks slot

• Assumptions: – Same click probability for every agent – Same valuation for every slot

GSP Auction

• Allocation: Highest bids win. Bidder gets slot .• Payment: Pay the bid of i+1

– No negative transfers • Per query payoff:

o.w.

GSP Properties

• , or else may lead to negative payoff– Individually rational

• Incentive Compatible?

• Exactly one slot => Vickery’ Second price auction

Locally Envy Free Equilibrium

• A bid profile such that no one wants to switch the slot with the person above.

• Existence of locally envy-free equilibrium with VCG payments

• Payment from locally envy-free equilibrium is at least as large as the payment from VCG (same bids)