Lecture 18: Incentives in BGP Routing and Review Announcements

• HW 8 is due tonight

• Requesting/uploading strategy notes:

• Clients send maxRequests to each peer (redundant requests are expected)

• If multiple “rare” pieces: randomize to break symmetry

• Any assumptions you make, you should put comments in the code along with mentioning it in the write up

• Project proposal google form:

• Only indicate partner status and anything you want me to know ahead of one-on-one meeting

• Reminder: Midterm 2 (24 hours, Nov 17), no class Transition to Remote/Project Work

• Class will transition from lecture-based to project-based

• Outline description of default project option (will release today): Understanding, implementing and analyzing strategies in two papers (BitTyrant and BitTorrent as an Auction (Prop Share)

• Project proposal google form:

• Due Sat (partner status, project (default or otherwise)

• In place of a lecture next Thursday, 20-30 min group meetings to discuss project plans, and set up meeting times for remote semester

• Look out for when2meet/doodle to indicate meeting availability

• Written (at most 500 word) project proposal due next Friday (Nov 20)

• All project materials: submitted through Github Remote Planning

• Mandatory meeting once a week to discuss project progress

• Second optional meeting

• We will schedule both just in case

• Focus is on final project

• Whether or not we do synchronous misc-lectures: depends on schedule of group meetings: if synchronous, it will be optional and recorded

• Last week likely to be somewhat synchronous

• Wrap up class/project presentations

• If you are in a later time zone, can record presentation

• Look out for all of these announcements/planning on Slack Incentives: Network Routing

• Last week we discussed incentives in P2P systems

• Today I want to talk about incentives when it comes to routing protocols in computer networks

• Two types of routing:

• Selfish routing in local area networks

• Braess paradox

• Price of anarchy

• Network over provisioning

• Inter-domain routing in the Internet Routing Between ASes

• How can you find a route from point A to B when they are located in different ASes

• Potentially requires traversing multiple ASes

• Different ASes may have different preferences:

• May want to minimize monetary cost of paths

• For example: consider the network given in figure

• Suppose d is destination for all traffic • ASes 1,2 have their preferences labelled

• Both prefer to route through the other than direct

Credit: Roughgarden Border Gateway Protocol (BGP)

• In the internet, routing between ASes is done using the Border Gateway Protocol (BGP)

• We will only discuss a sketch of how it works

• Ignore many details of the actual protocol

• Fix a destination d (BGP runs in parallel for all choices of d) • Each destination d broadcasts their presence to neighbor ASes • Each AS is then supposed to update their own path to d and broadcast to their neighboring ASes

• All messages are asynchronous BGP Updates

• At an AS u with neighbors N • For each v ∈ N

• Let be the last path (from to ) that announced to Pv v d v u (if no path then is empty)

• Reset to 's favorite cycle-free path of the form Pu u Note that this is the "intended (if any, otherwise empty); is concatenation (u, v) ∘ Pv ⋅ behavior": of course ASes may not follow it: more on that in a bit • If changes, announce the new value to all neighbors Pu v ∈ N

• Note that AS has to avoid cycles: if includes , cannot route u Pv u traffic through Pv BGP Example

• Multiple possible fixed points of running BGP

• Which fixed point do we expect to reach?

• Depends on timing of messages: whichever 1 or 2 finds out first that the other is using a direct path to d can switch and "win"

Credit: Roughgarden Stable Routings: Equilibrium

• In a stable routing, no AS wants to change its path to d, given the choices of other ASes and options available to u • This is a Nash equilibrium in a game where

• Players: AS, available strategies: neighboring ASes, payoffs induced by preferences over paths

Credit: Roughgarden Stable Routings: Equilibrium

• Note: a stable routing must a tree, directed into the destination d • BGP maintains the invariant that out-degree of every AS is at most 1 • BGP also explicitly prevents cycles from forming

• We saw that there can be multiple stable routings

• Question. Does a stable routing always exist?

Credit: Roughgarden Stable Routings: May Not Exist

• Consider the following stable routing network

• Every AS prefers a direct path to d over the empty path • Thus, one of the ASes must have a direct path to d

Credit: Roughgarden Dispute Wheels

• We want to discuss incentives in BGP routing

• Do ASes have an incentive to follow the intended behavior

• Hard to do if the output of BGP is not well defined in some cases

• Under what cases does the output of BGP make sense?

• Theorem [Griffin et al.] If the AS graph has no dispute wheel, then:

• There exists a stable routing

• This stable routing is unique

• BGP is guaranteed to converge to the stable routing

• What is a dispute wheel? Dispute Wheels

• A dispute wheel consists of (for some k ≥ 2):

• distinct ASes u1, u2, …, uk

• cycle-free paths where is a path P1, P2, …, Pk Pi from to ui d

• cycle-free paths , where is a path Q1, Q2, …, Qk Qi from to ui ui+1 such that for each , prefers the indirect i = 1,…, k ui path to the direct path (here Qi ⋅ Pi+1 Pi ) uk+1 = u1, Pk+1 = P1

• Note that 's and need not be internally disjoint Pi Qi′s

Credit: Roughgarden Dispute Wheels Example

• , P1 = 1d P2 = 2d

• Q1 = 12, Q2 = 21 • Both ASes prefer 12d or 21d rather than direct

Credit: Roughgarden Dispute Wheels Example

• for Pi = id i = 1,2,3

• Q1 = 12, Q2 = 23, Q3 = 31 • Again each AS prefers indirect over direct

Credit: Roughgarden Dispute Wheels in Practice

• BGP converges if there are no dispute wheels

• But how strong is the no-dispute-wheel condition?

• Gao and Rexford gave justifications about why no dispute wheel condition should generally hold for realistic AS preferences

• Gao-Rexford conditions (rule out dispute wheels): every AS prefers

• to route through a customer over those through a peer

• to route through a peer over a provider

• Empirical evidence the conditions are approximately true for most ASes Incentive Issues

• We restrict ourselves to AS graphs where BGP converges

• Do ASes have an incentive to follow the protocol?

• Does any AS have a beneficial unilateral deviation?

• Types of deviations:

• Choose your path to be something other than the favorite path Pu among the available options

• Withhold information about your path to (some) neighbors

• Announce a path to (some) neighbors that is different from the one you are actually using, possible even a non-existent fake path Fake Path Announcements

• Can and do ASes announce non-existent paths?

• Happens in BGP all the time

• In 2008 Pakistan telecom blocked access from Pakistan to Youtube

Credit: Roughgarden Fake Path Announcements

• In 2008 Pakistan telecom blocked access from Pakistan to Youtube

• Pakistan Telecom rerouted Youtube traffic to a local server

• Pakiston Telecom announced this new route to Youtube to some of its neighbors

• An ISP in HK switched its route to Youtube soon, followed by most of the other ISPs

• Outcome: a lot of users could not reach Youtube, Pakistan Telecom inundated with requests

Credit: Roughgarden Withholding Path Information

• Withholding deviations from neighboring AS can be beneficial

• If AS 1 never announces any paths and AS 2 follows BGP, then the protocol converges to the stable routing preferred by AS 1

• What if there are no dispute wheels (this example does have them)

Credit: Roughgarden Announcing Fake Paths

• If edge 3d was in the network and AS 3 preferred this path less than 31d, the example would have a dispute wheel • Suppose all ASes are honest, since there is no dispute wheel, BGP converges to a unique stable routing: shown in (a)

Credit: Roughgarden Announcing Fake Paths

• No suppose AS 3 announces the fake path 3d to its neighbors • Creates a fictitious dispute wheel

• BGP converges to figure (b): AS 2 thinks its traffic is being routed through 23d, when in fact it is being routed through 31d

Credit: Roughgarden Announcing Fake Paths

• If AS 2 knew the path wasn’t being routed through 23d: would not be stable but without that, it is

• AS 3 gets its favorite path when announcing a fake route, compared to being truthful

Credit: Roughgarden BGP with Path Verification

• While fake path announcements are possible and frequent in current BGP protocol, there is work being done to eliminate it

• BGPsec protocol uses cryptographic signatures to verify the existence of announced paths

• Suppose we are able to detect and disallow fake path announcements

• Question. Is BGP then incentive compatible?

Credit: Roughgarden BGP with Path Verification

• Theorem. (Levin et al.). Assuming that no AS can announce a path that doesn’t exist and the AS graph does not have a dispute wheel

• No AS has an incentive to unilaterally deviate from the intended behavior in BGP (assuming other ASes follow BGP)

Credit: Roughgarden http://ccr.sigcomm.org/online/files/p267-goldberg.pdf http://cs.brown.edu/courses/csci2950-g/papers/ic-routing.pdf Midterm Review Overview of topics

• Basic game theory

• Normal-form games, dominant strategy, Nash equilibria, mixed-Nash equilibria

• Introduction to auctions

• DSIC auctions, first-price and second-price

• Bayes Nash equilibrium to analyze first-price auctions

• Myerson’s lemma and Revenue equivalence

• Advanced auction theory:

• VCG auctions for surplus maximization

• Revenue maximization using reserve prices Overview of topics

• Sponsored search auctions

• GSP auction and balanced bidding

• Matching markets

• Markets with money: competitive equilibrium

• Markets without money: stable matching, serial dictatorship and TTC

• Voting and social choice theory

• Extensive form games, repeated games and incentives in P2P

• Incentives in Network routing (Braess paradox/BGP) Pre Midterm 1 Dominant Strategy Equilibrium

• Domination: For a player an action (weakly) dominates action if i, ai a′ ∈ A it is always beneficial to play a over a′

• That is, for all : a−i ∈ A−i ui(ai, a−i) ≥ ui(ai′, a−i) and the inequality is strict for some a−i ∈ A−i

• If the inequality is strict for all then we have (strong) domination a−i

• An action is dominant for player if it weakly dominates all actions a ∈ Ai i

a′ ≠ a ∈ Ai • Dominant strategy equilibrium (DSE)

• An equilibrium where each player plays their dominant action Nash Equilibrium

• A profile of actions is a pure strategy Nash equilibrium if a = (a1, …, an) ∈ A

for each player and for all i ai′ ∈ Ai

ui(ai, a−i) ≥ ui(ai′, a−i)

• In Nash equilibrium, players play their best response to the "given strategy" of others (need to reason about what other players are doing)

• More general concept than DSE; weaker guarantee than DSE

• A dominant strategy equilibrium is always a Nash equilibrium but a Nash equilibrium may not be a dominant strategy equilibrium Pareto Optimality

• An outcome o is at least as good for every player, as another outcome o′, and

there is some agent who strictly prefers o to o′

• Reasonable to say o is better than o′

• We say o Pareto dominates o′

• Definition (Pareto Optimality).

• An outcome o* is Pareto-optimal if there is no other outcome that Pareto dominates it Mixed Strategy and Mixed Nash

• Mixed Strategy. A mixed strategy for agent , with si : Ai → [0,1] i assigns a probability to each action such that si ∈ Δ(Ai) pij j ∈ Ai

∑ pij = 1 j∈Ai

• Best response definition

• BR iff si* ∈ (s−i) ui(si*, s−i) ≥ ui(si, s−i) ∀si ∈ Si

• Nash equilibrium definition

• A strategy profile is a mixed-strategy Nash s = (s1, …, sn) ∈ S equilibrium iff BR ∀i si ∈ (s−i) Complexity of Nash

• Two-player, zero-sum games are special

• Can efficiently compute mixed Nash using LP duality

• In general, finding mixed-Nash is PPAD hard

�� Complete

�� Theorem [Daskalakis, Goldberg, Papadimitriou ’06]. PPAD FindNash problem is PPAD-complete � Auctions

• Bidder valuations is their private information: games of incomplete information • Single-item sealed bid auction:

• Each bidder has private value for the item i vi ∈ [0,vmax] • Each bidder submits a bid bi ≥ 0 • Given bid profile : b = (b1, …bn) • An allocation rule x(b) ∈ {0,1}n, indicates whether bidder i receives the item or not, i.e. or xi(b) = 1 0 • A payment rule n, specifies the payment bidder must make p(b) ∈ ℝ pi(b) i • Quasi-linear utility. Given a bid profile b, the utility of bidder i for the allocation rule and payment is xi(b) ∈ {0,1} ti(b) ui(b) = xi(b) ⋅ vi − pi(b) • If a bidder wins item and pays then its utility is p vi − p • If a bidder loses item and pays nothing, its utility is 0 Second-Price & First-Price Auctions

• Vickrey auction: bidder with highest bid wins, pays second highest bid • Vickey auction is awesome: • Maximizes social welfare/surplus • DSIC • Polynomial time • First price auction: highest bidder wins and pays their bid • To analyze them, we needed Bayes Nash equilibrium

• A strategy profile is a pure strategy Bayes Nash equilibrium if no s = (si, s−i) player can increase their interim (or ex-ante) expected utility by unilaterally changing their strategy Uniform Distribution

• Models situations where all outcomes in the range have equal probability • Probability density function of a continuous uniform distribution on [a, b]

• Cumulative density function of a continuous uniform distribution on [a, b] First-Price Auction: n Bidders

• Suppose we increase the number of bidders, how should the equilibrium strategy adjust to more competition?

• Theorem. Assume each of the n bidders have values drawn i.i.d. from uniform distribution on [0,1]. Then, the n − 1 strategy s(v ) = ⋅ v is a symmetric Bayes Nash i n i equilibrium of the sealed-bid first price auction.

• Proof. We can generalize the 2-bidder proof

• On board. Also in Parkes and Seuven book.

• Takeaway: the more the competition, the more the bidders need to bid closer to their value if they want to win Order Statistics

• We want to compare the expected revenue of first-price and second-price auctions • To do so, we need to define order statistics

• Let be independent samples drawn X1, X2, …Xn n identically from the uniform distribution on [0,1] The first-order statistic is the maximum value of the • X(1) samples, the second-order statistic is the second- X(2) maximum value of the samples, etc • The expected value of the kth order statistic for n i.i.d samples from U(0,1) is

n − (k − 1) E[X ] = (k) n + 1

Expected kth order statistic for 3 samples, uniform [0,1] Myerson’s Lemma

• Fix an single-parameter domain. We state the result for the continuous case. Myerson’s Lemma. (a) An allocation rule x can be made dominant-strategy incentive compatible if and only if x is monotone (non decreasing). (b) If x is monotone, there is a unique payment rule p such that (x, p) is DSIC. This payment rule is given by the following expression for all i: vi

pi(v) = vi ⋅ xi(vi, v−i) − xi(z, v−i) dz ∫0 Keeping fixed, we can simplify it as: v−i vi

pi(vi) = vi ⋅ xi(vi) − xi(z) dz ∫0 Assuming that . pi(0) = 0

Credit: Hartline’s Book on Mechanism Design Myerson’s Lemma: HW 3 Problem 1

• Surplus maximizing allocation:

• Sort and reindex bids so that: b1β1 ≥ b2β2 ≥ …bnβn • Assign slot 1 to bidder 1, 2 to bidder 2 and so on • Allocation rule is monotone:

• Bidder 's quality stays the same, increasing bid can only i βi ≥ 0 increase bidder's ranking in the greedy allocation • For payment rule we apply Myerson's lemma:

• Consider bidder and , fix i bi b−i • Allocation curve is a step-wise graph and the payment rule is bi jump in at pi(bi) = ∑ zj ⋅ xi zj i=1 Myerson’s Lemma: HW 3 Problem 1

• For payment rule we apply Myerson's lemma:

• Consider bidder and , fix i bi b−i • Allocation curve is a step-wise graph and the payment rule is

bi p (b ) = z ⋅ jump in x at z i i ∑ j i j i=1

• To find the critical bids at which the jump occurs, lets increase bi from , keeping fixed 0 b−i • 's allocation jumps from getting no clicks to clicks when i βiαk that is biβi = βkbk

βk bi = bk βi Myerson’s Lemma: HW 3 Problem 1

• To find the critical bids at which the jump occurs, lets increase from bi , keeping fixed 0 b−i • 's allocation jumps from getting no clicks to clicks when i βiαk that is biβi = βkbk

β k bi = bk βi βj Similarly for slot , the critical bid is • j ⋅ αj βi Size of jump in allocation is: • αjβi − αj+1βi = βi(αj − αj+1) • Thus total payment is: k k βj ∑ bj ⋅ βi(αj − αj+1) = ∑ bjβj ⋅ (αj − αj+1) j=i+1 βi j=i+1 Revenue Equivalence

• Two auctions with the same allocation rule, and the same distributional assumptions on players generate the same revenue at a BNE

• Step 1. Guess what the allocation might be in a Bayes-Nash equilibrium (usually a surplus-maximizing one)

• Step 2. Calculate the interim expected payment of an agent in terms of the strategy function

• Step 3. Calculate the interim expected payment of an agent in a strategically-simpler auction (usually the second-price version)

• Step 4. Solve for the BNE strategy by setting the expected payment equal to each other

• Step 5. Finally, verify that the initial guess was correct and the strategy is a symmetric BNE of the mechanism VCG Mechanism

• Given bids where each is now a vector indexed b = (b1, b2, …, bn) bi by |Ω| , return the surplus maximizing allocation (assuming bids as proxies for valuations): n argmax x(b) = ω∈Ω ∑ bi(w) i=1 • Payment rule given by

argmax bj(w) bj(ω*) pi(b) = w∈Ω−i ∑ − ∑ j≠i j≠i

without i with i Where ω* is the surplus maximizing outcome in the presence of i VCG Mechanism

• Given bids where each is now a vector indexed b = (b1, b2, …, bn) bi by |Ω| , return the surplus maximizing allocation (assuming bids as proxies for valuations): n argmax x(b) = ω∈Ω ∑ bi(w) i=1 • Payment rule given by

argmax bj(w) bj(ω*) pi(b) = w∈Ω−i ∑ − ∑ j≠i j≠i

without i with i Where ω* is the surplus maximizing outcome in the presence of i Post Midterm 1 Revenue Maximization Virtual value can be negative! • Consider single item, uniform iid bidders from (0,1)

• The virtual valuation of a bidder with valuation is defined as ϕ(v) vi

1 − F(vi) ϕ(vi) = vi − = 2vi − 1 f(vi)

• Remember the result: 1 v 1/2 Expected Revenue = Expected Virtual Surplus −1

Virtual value is strictly increasing with value Myerson Optimal Auction

• Allocation rule:

• Accept bids and compute virtual values , b = (b1, …, bn) ϕ(b1) ,.., ϕ(b2) ϕ(bn)

• Sort and reindex so that ϕ(b1) ≥ ϕ(b2) ≥ …ϕ(bn)

• Award the good to the person iff , else no sale 1 ϕ(b1) ≥ 0 • Payment rule: what do they pay?

• Critical bid: the lowest bid b* that the bidder could have made and still gotten the item: ϕ(b*) = max{ϕ(b2),0}

• If : sell at price ϕ(b2) ≥ 0 b* = b2 • Else, sell at price b* = ϕ−1(0) (the bid at which ϕ(b*) = 0) Vickrey Auction with Reserve

• Let r = ϕ−1(0), then we our revenue-optimal auction becomes: • Ignore bids less than r, award to highest remaining bidder • Charge payment that is max{second highest bid, r} • It is incredible that a complicated question like this, led to a simple clean answer like the Vickrey auction with a reserve price!

This is just the Vickrey auction with a suitable reserve! The Power of One Additional Bidder

• Implies Vickrey auction always generates (n − 1)/n times the revenue of the optimal auction!

• One additional bidder has the same power as if we knew the distribution of bidder values ahead of time!

• This result that has anecdotal support in practice:

• Extra competition is more important than getting the details of the auction just right!

• More useful to generate interest from more participants, than learning more about their preferences!

[Bulow Klemperer] Vickrey auction (with no reserve) with n + 1 bidders generates just as much expected revenue as the revenue-optimal auction with n bidders! Sponsored Search Auction

• We have k ad slots and n bidders (advertisers)

• Each add slot has a "click through rate" that models the number of i αi clicks generated by the slot

• We assume that α1 ≥ α2 ≥ … ≥ αk • Assumption. No quality parameter for now (but everything generalizes)

• Bidders have a private value per click vi Bidder 's value for being assigned slot is • i j vi ⋅ αj

• Each bidder submits a "bid-per-click" (amount they are willing to pay per click)

• Auction collects the bid profile and outputs an allocation b = (b1, …, bn) rule x(b) (who gets what slot) and payment rule p(b) (who pays what) Truthful VCG Mechanism

• Dominant strategy incentive compatible mechanism

• Surplus maximizing allocation:

• Rank bidders by the bid value b1 ≥ b2 ≥ … ≥ bn • Assign slot i to bidder i, where 1 ≤ i ≤ k • Payment rule: k • Total payment: ∑ bj+1 ⋅ (αj − αj+1) j=i

1 k • Per click payment: ∑ bj+1 ⋅ (αj − αj+1) αi j=i Generalized Second Price Auction

• Collect bids-per-click b = (b1, …, bn) • GSP allocation rule: same as VCG

• The price-per-click for bidder who is assigned slot i:

• "Critical bid": just the bid of the person below bi+1 i

• Envy-free outcome. We say that a bid profile where b = (b1, …, bn) is envy-free if for every bidder b1 ≥ b2 ≥ … ≥ bn i

αi(vi − bi+1) ≥ αj(vi − bj+1)

• Lemma. An envy-free outcome of the GSP auction must be a Nash equilibrium Balanced Bidding

• We say a bid profile satisfies the balanced bidding b = (b1, b2, …, bn) requirement if • The following holds for bidder i for 2 ≤ i ≤ m

αi(vi − bi+1) αi−1(vi − bi)

utility current position = utility in case of retaliation

• Any unassigned bidder bids their true value • Lemma. There exists an envy-free Nash equilibrium of the GSP auction where all bidder's bids are value ordered and satisfy balanced bidding condition.

Credit: Textbook by Parkes and Seuken Bigger Picture

GSP

Nash eq Envy free BB = VCG

Figure adapted from Textbook by Parkes and Seuken One-Sided Matching Markets

• Each buyer wants only one item & each item can be given to at most one buyer

• n potential buyers, m ≥ n items (if this is not true, we can always create dummy items that everyone values at 0) Each buyer has a private valuation for each item • i vij ≥ 0 j If the price of item is , the utility that person receives from • j pj i getting item j is uij = vij − pj

• Preferred-item graph (given prices p): nodes are items and buyers and there is an edge between buyers and their preferred items (items that give them max utility) Market-Clearing Prices

• A selection of prices is market-clearing if: p = (p1, p2, …, pm) • Condition 1. There is a matching in the preferred-item graph that such that all buyers are matched to an item

• Condition 2. If an item j is not matched to any buyer, then its price , in other words, every item with non-zero price pj = 0 must get sold pj > 0 • This means that at market-clearing prices, each buyer can come by and pick some item that maximizes its utility

• Assume tie-breaks can occur in a coordinated way

• Furthermore, at these prices the market will "clear"

• Only items left behind are those that are not desirable (price 0) First Welfare Theorem

• Matchings in a competitive equilibrium are exactly the matching with maximum possible value

• First Welfare Theorem. If (M, p) is a competitive equilibrium, then M is a matching with maximum total value, that is, n n

for every matching v ≥ v M′ • ∑ iM(i) ∑ iM′(i) i=1 i=1

• In particular, among all possible ways of allocating items such that each buyer is matched to at most one item good and each item is matched to at most one buyer, the allocation achieved at a competitive equilibrium maximizes welfare Competitive Eq: Existence

• Theorem. In every market where at most one good is assigned to each buyer, there is at least one competitive equilibrium

• Equivalently, market-clearing prices are guaranteed to exist

• We prove this constructively through an mechanism that shows how such prices might emerge organically in a market

• Intuition idea behind our "ascending-price mechanism"

• If a set of k items is preferred by more than k buyers at its current price, then the prices of these items should rise

• Keep identifying such "constricted sets" and increasing prices until the market clears VCG Prices are Market Clearing

• Despite their definition as personalized prices, VCG prices are always market clearing (for the case when each buyer wants a single item)

• Suppose we computed VCG prices for a given matching market

• Then, instead of assigning the VCG allocation and charging the price, we post the prices publicly

• Without requiring buyers to follow the VCG match

• Despite this freedom, each buyer will in fact achieve the highest utility by selecting the item that was allocated by the VCG mechanism!

• Claim. In any matching market (where each buyer can receive a Since GSP prices are identical to VCG single item) the VCG prices form the unique set of market clearing at a balanced-bidding envy-free equilibrium, they are also market clearing prices of minimum total sum. Stable Matching

• A set H of n hospitals, a set S of n students and their preferences (each hospital ranks each student, each students ranks each hospital)

• A hospital h and student s form a blocking pair (h, s) ∈ H × S in a matching M if • h prefers s to its current match in M

• s prefers h to its current match in M

• A matching M is stable if it does not contain a blocking pair.

• The Gale Shapely (deferred acceptance) algorithm produces a stable matching which is optimal for the proposing side. Stable Matching

• The side of the market that receives offers has an incentive to misreport and get a better matching

• Not strategy proof

• Theorem. No mechanism for two-sided matching is both stable and strategyproof. Serial Dictatorship

• We have n agents and n items • Agents have strict preference ordering over the items

• Care only about their own allocation, not others

• Each of the n agents submit a ranked ordering over items • Each agent is assigned a rank from {1,2,…, n} • For i = 1,2,…, n • Agent i is assigned their favorite choice among options still available

• Lemma. The serial dictatorship mechanism is strategyproof and Pareto optimal. Top Trading Cycle

• There are n agents and each initially owns a house • Each agent has a total ordering over the n houses • Each agent report their overall preferences in the beginning

• Step 1. Each agent (simultaneously) points to its favorite house (among houses remaining)

• Pick directed cycles and make all trades on it (each agent gives its house to the agent that points to it)

• Delete all agents and houses that were traded in Step 1

• While agents remain, go back to Step 1.

• Lemma. TTC mechanism is DSIC and produces a unique core allocation. Voting

• A set A of alternatives and a set N = {1,2,…, n} of voters • Each agent i ∈ N has a strict preference order alternatives: given by a ranked list and submits this to the mechanism Li • Voting rules can have two forms: either output is a single alternative (social choice function) or a full ranked list (social ranking function)

• Condorcet winner: an alternative that defeats every other alternative (a beats b if a majority of voters prefer a to b)

• May not exist (Condorcet paradox)

• A social choice function f satisfies the Condorcet criterion (is Condorcet consistent) if whenever there is a Condorcet winner, then f selects this alternative Voting

• Borda count: voters submit their full ranked lists: an alternate gets |A| for each 1st-choice vote, |A| − 1 points for each 2nd- choice vote, and so on and 1 point for each last-choice vote • For each vote, a positional-scoring rule on m = |A| alternatives assigns a score of to the alternative ranked in th αj j place. The alternative with maximum total score (across all votes) is selected.

• Assume and α1 ≥ α2 ≥ …αm α1 > αm • Examples: plurality, veto, Borda count

• Plurality, veto and Borda are not Condorcet consistent

• Plurality and Borda do not satisfy independence of clones Ranked-Choice Voting

• Alternative to plurality: also called single-transferable vote (STV) or instant-runoff voting

• Voters submit a full ranked list (not just their first choice)

• If there is some alternative a* that receives more than 50% of the first-place voters, then a* is the winner

• Otherwise, the alternative with the fewest first-place votes is deleted and the winner is computed recursively on the rest

• Base case: only two alternatives left, use majority rule

• Notice that this rule is not biased towards "extreme candidates"

• Various tie-breaking rules used in case of ties Impossibility Results

• Gibbard-Satterthwaite theorem. Any voting rule with at least 3 alternatives that is strategyproof and onto must be dictatorial.

• Arrow's impossibility theorem. With three or more alternatives, no social-rank function satisfies the following three properties:

• Non-dictatorship

• Unanimity

• Independence of irrelevant alternatives (IIA)

• A voter i has single-peaked preferences if there is a “peak” such that the voters prefers candidates closer to her peak pi ∈ ℝ • Median rule is Pareto optimal and satisfies the Condorcet criterion (will discuss this soon) under single-peaked preferences Acknowledgments

• These lecture slides are partly based off the notes by Tim Roughgarden:

• http://timroughgarden.org/f16/l/l8.pdf

Credit: Roughgarden