1. Social choice theory = preference aggregation = voting assuming agents tell the truth about their preferences
Tuomas Sandholm
Professor
Computer Science Department
Carnegie Mellon University

2. Social choice Collectively choosing among outcomes
E.g. presidents
Outcome can also be a vector
E.g. allocation of money, goods, tasks, and resources
Agents have preferences over outcomes
Center knows each agent’s preferences
Or agents reveal them truthfully by assumption
Social choice function aggregates those preferences & picks outcome
Outcome is enforced on all agents
CS applications: Multiagent planning [Ephrati&Rosenschein], computerized elections [Cranor&Cytron], accepting a joint project, collaborative filtering, rating Web articles [Avery,Resnick&Zeckhauser], rating CDs...

3. Condorcet paradox [year 1785]
Majority rule
Three voters:
x > z > y
y > x >z
z > y > x
x > z > y > x
Unlike in the example above, under some preferences there is a Condorcet winner,
i.e., a candidate who would win a two-candidate election against each of the other candidates.

4. Agenda paradox x y z x y z x y z Binary protocol (majority rule) = cup
Three types of agents:
x > z > y (35%)
y > x > z (33%)
z > y > x (32%) x y z x y z x y z
Power of agenda setter (e.g. chairman)
Vulnerable to irrelevant alternatives (z)
Plurality protocol
For each agent, most preferred outcome gets 1 vote
Would result in x

5. Pareto dominated winner paradox
Voters:
x > y > b > a
a > x > y > b
b > a > x > y

6. Inverted-order paradox
Borda rule with 4 alternatives
Each agent gives 4 points to best option, 3 to second best...
Agents:
x=22, a=17, b=16, c=15
Remove x: c=15, b=14, a=13
x > c > b > a
a > x > c > b
b > a > x > c

7. Borda rule also vulnerable to irrelevant alternatives
Three types of agents:
Borda winner is x
Remove z: Borda winner is y
x > z > y (35%)
y > x > z (33%)
z > y > x (32%)

8. Majority-winner paradox
Agents:
Majority rule with any binary protocol: a
Borda protocol: b=16, a=15, c=11
a > b > c
b > c > a
b > a > c
c > a > b

9. Is there a desirable way to aggregate agents’ preferences?
Set of alternatives A
Each agent i in {1,..,n} has a complete, transitive (not necessarily strict) ranking <i of A
Complete, transitive (not necessarily strict) social welfare function F: Ln -> L
Some (weak) desiderata of F
1. Unanimity: If all voters have the same ranking, then the aggregate ranking equals that. Formally, for all < in L, F(< ,…,<) =<.
2. Nondictatorship: No voter is a dictator. Voter i is a dictator if for all <1 ,…,<n , F(<1 ,…,<n) = <i
3. Independence of irrelevant alternatives: The social preference between any alternatives a and b only depends on the voters’ preferences between a and b. Formally, for every a, b in A and every <1 ,…,<n , < ’1 ,…,< ’n in L , if we denote < = F(<1 ,…,<n) and < ’ = F(< ’1 ,…,< ’n), then (a <i b <=> a < ’i b for all i) implies that (a < b <=> a < ’ b).
Arrow’s impossibility theorem [1951]: If |A| ≥ 3, then no F satisfies desiderata 1-3.

10. Proof (from “A One-shot Proof of Arrow's Impossibility Theorem”, by Ning Neil Yu)
Because k can be anywhere
in the others’ preferences once
we ignore i.

11. Stronger version of Arrow’s theorem
In Arrow’s theorem, social welfare function F outputs a ranking of the outcomes
The impossibility holds even if only the highest ranked outcome is sought:
Thm. Let |A| ≥ 3. If a social choice function f: Ln -> A is monotonic and Paretian, then f is dictatorial.
Definition. f is monotonic if [ x = f(>) and x maintains its position in >’ ] => f(>’) = x
Definition. x maintains its position whenever x >i y => x >i’ y
Proof. From f we construct a social welfare function F that satisfies the conditions of Arrow’s theorem
For each pair x, y of outcomes in turn, to determine whether x > y in F, move x and y to the top of each voter’s preference order
don’t change their relative order
(order of other alternatives is arbitrary)
Lemma 1. If any two preference profiles >’ and >’’ are constructed from a preference profile > by moving some set X of outcomes to the top in this way, then f(>’) = f(>’’)
Proof. Because f is Paretian, f(>’)  X. Thus f(>’) maintains its position in going from >’ to >’’. Then, by monotonicity of f, we have f(>’) = f(>’’)
Note: Because f is Paretian, we have f = x or f = y (and, by Lemma 1, not both)
F is transitive (total order) (we omit proving this part)
F is Paretian (if everyone prefers x over y, then x gets chosen and vice versa)
F satisfies independence of irrelevant alternatives (immediate from Lemma 1)
By earlier version of the impossibility, F (and thus f) must be dictatorial. ■

12. Voting rules that avoid Arrow’s impossibility (by changing what the voters can express)
Approval voting
Each voter gets to specify which alternatives he/she approves
The alternative with the largest number of approvals wins
Avoids Arrow’s impossibility
Unanimity
Nondictatorial
Independent of irrelevant alternatives
Range voting
Instead of submitting a ranking of the alternatives, each voter gets to assign a value (from a given range) to each alternative
The alternative with the highest sum of values wins
Independent of irrelevant alternatives (one intuition: one can assign a value to an alternative without changing the value of other alternatives)
More information about range voting available at
These still fall prey to strategic voting (e.g., Gibbard-Satterthwaite impossibility, discussed in the next lecture)

13. Mechanism design (strategic “voting”)
Tuomas Sandholm
Professor
Computer Science Department
Carnegie Mellon University

14. Goal of mechanism design
Implementing a social choice function f(R) using a game
Actually, say we want to implement f(u1, …, u|A|)
Center = “auctioneer” does not know the agents’ preferences
Agents may lie
unlike in the theory of social choice which we discussed in class before
Goal is to design the rules of the game (aka mechanism) so that in equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)
Mechanism designer specifies the strategy sets Si and how outcome is determined as a function of (s1, …, s|A|)  (S1, …, S|A|)
Variants
Strongest: There exists exactly one equilibrium. Its outcome is f(u1, …, u|A|)
Medium: In every equilibrium the outcome is f(u1, …, u|A|)
Weakest: In at least one equilibrium the outcome is f(u1, …, u|A|)

15. Revelation principle
Any outcome that can be supported in Nash (dominant strategy) equilibrium via a complex “indirect” mechanism can be supported in Nash (dominant strategy) equilibrium via a “direct” mechanism where agents reveal their types truthfully in a single step
Agent 1’ s
preferences
Agent |A|’ .
Strategy
formulator
Original
“complex”
“indirect”
mechanism
Outcome
Constructed “direct revelation” mechanism

16. Uses of the revelation principle
Literal: “Only direct mechanisms needed”
Problems:
Strategy formulator might be complex
Complex to determine and/or execute best-response strategy
Computational burden is pushed on the center (i.e., assumed away)
Thus the revelation principle might not hold in practice if these computational problems are hard
This problem traditionally ignored in game theory
Even if the indirect mechanism has a unique equilibrium, the direct mechanism can have additional bad equilibria
As an analysis tool
Best direct mechanism gives tight upper bound on how well any indirect mechanism can do
Space of direct mechanisms is smaller than that of indirect ones
One can analyze all direct mechanisms & pick best one
Thus one can know when one has designed an optimal indirect mechanism (when it is as good as the best direct one)

17. Implementation in dominant strategies
A “strong” form of mechanism design

18. Implementation in dominant strategies
Goal is to design the rules of the game (aka mechanism) so that in dominant strategy equilibrium (s1, …, s|A|), the outcome of the game is f(u1, …, u|A|)
Nice in that agents cannot benefit from counterspeculating each other
Others’ preferences
Others’ rationality
Other’s endowments
Other’s capabilities …

19. Gibbard-Satterthwaite impossibility
Thrm. If |O | ≥ 3 (and each outcome would be the social choice under f for some input profile (u1, …, u|A|) ) and f is implementable in dominant strategies, then f is dictatorial
Proof. (Assume for simplicity that utility relations are strict)
By the revelation principle, if f is implementable in dominant strategies, it is truthfully implementable in dominant strategies with a direct revelation mechanism
Since f is truthfully implementable in dominant strategies, the following holds for each agent i: ui(f(ui,u-i)) ≥ ui(f(ui’,u-i)) for all u-i
Claim: f is monotonic. Proof: Suppose not. Then there exists u and u’ s.t. f(u) = x, x maintains position going from u to u’, and f(u’)  x
Consider converting u to u’ one agent at a time. The social choices in this sequence are, e.g., x, x, y, …, z. Consider the first step in this sequence where the social choice changes. Call the agent that changed his preferences agent i, and call the new social choice y. For the mechanism to be truth-dominant, i’s dominant strategy should be to tell the truth no matter what others reveal. So, truth telling should be dominant even if the rest of the sequence did not occur.
Case 1. u’i(x) > u’i(y). Say that u’i is the agent’s truthful preference. Agent i would do better by revealing ui instead (x would get chosen instead of y). This contradicts truth-dominance.
Case 2. u’i(x) < u’i(y). Because x maintains position from ui to u’i, we have ui(x) < ui(y). Say that ui is the agent’s truthful preference. Agent i would do better by revealing u’i instead (y would get chosen instead of x). This contradicts truth-dominance.
Claim: f is Paretian. Proof: Suppose not. Then for some preference profile u we have an outcome x such that for each agent i, ui(x) > ui(f(u)).
We also know that there exists a u’ s.t. f(u’) = x
Now, choose a u’’ s.t. for all i, ui’’(x) > ui’’(f(u)) > ui’’(z), for all z  f(u), x
Since f(u’) = x, monotonicity implies f(u’’) = x (because going from u’ to u’’, x maintains its position)
Monotonicity also implies f(u’’) = f(u) (because going from u to u’’, f(u) maintains its position)
But f(u’’) = x and f(u’’) = f(u) yields a contradiction because x  f(u)
Since f is monotonic & Paretian, by strong form of Arrow’s theorem, f is dictatorial. ■

20. Ways around the Gibbard-Satterthwaite impossibility
Use a weaker equilibrium notion
E.g., Bayes-Nash equilibrium
In practice, agent might not know others’ revelations
Design mechanisms where computing a beneficial manipulation (insincere ranking of outcomes) is hard
NP-complete in second order Copeland voting mechanism [Bartholdi, Tovey, Trick 1989]
Copeland score: Number of competitors an outcome beats in pairwise competitions
2nd order Copeland: Copeland, and break ties based on the sum of the Copeland scores of the competitors that the outcome beat
NP-complete in Single Transferable Vote mechanism [Bartholdi & Orlin 1991]
NP-hard, #P-hard, or PSPACE-hard in many voting protocols if one round of pairwise elimination is used before running the protocol [Conitzer & Sandholm IJCAI-03]
Weighted coalitional manipulation (and thus unweighted individual manipulation when the manipulator has correlated uncertainty about others) is NP-complete in many voting protocols, even for a constant #candidates [Conitzer, Sandholm & Lang JACM 2007]
“Typical case” complexity tends to be easy [Conitzer&Sandholm AAAI-06, Procaccia&Rosenschein JAIR-07, Friedgut, Kalai&Nisan FOCS-08, Isaksson, Kindler&Mossel FOCS-10]
Randomization
Agents’ preferences have special structure
IC => convex combination of
(some randomization to pick a dictator)
and
(some randomization to pick 2 alternatives)
[Gibbard Econometrica-77]
General preferences
Quasilinear preferences

21. Quasilinear preferences: Groves mechanism
Outcome (x1, x2, ..., xk, m1, m2, ..., m|A| )
Quasilinear preferences: ui(x, m) = mi + vi(x1, x2, ..., xk)
Utilitarian setting: Social welfare maximizing choice
Outcome s(v1, v2, ..., v|A|) = maxx i vi(x1, x2, ..., xk)
Thrm. Assume every agent’s utility function is quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies if mi(v)= ji vj(s(v)) + hi(v-i) for arbitrary function h
Proof. We show that every agent’s (weakly) dominant strategy is to reveal the truth in this direct revelation (Groves) mechanism
Let v be agents’ revealed preferences where agent i tells the truth
Let v’ have the same revealed preferences for other agents, but i lies
Suppose agent i benefits from the lie: vi(s(v’)) + mi(v’) > vi(s(v)) + mi(v)
That is, vi(s(v’)) + ji vj(s(v’)) + h i(v-i’) > vi(s(v)) + ji vj(s(v)) + h i(v-i)
Because v-i’ = v-i we have h i(v-i’) = h i(v-i)
Thus we must have vi(s(v’)) + ji vj(s(v’)) > vi(s(v)) + ji vj(s(v))
We can rewrite this as j vj(s(v’)) > j vj(s(v))
But this contradicts the definition of s() ■

22. Uniqueness of Groves mechanism
Thrm. Assume every agent’s utility function is quasilinear. A utilitarian social choice function f: v -> (s(v), m(v)) can be implemented in dominant strategies for all v: A x O -> R only if mi(v)= ji vj(s(v)) + hi(v-i) for some function h
Proof.
Wlog we can write mi(v) = ji vj(s(v)) + hi(vi , v-i)
We prove hi(vi , v-i) = hi(v-i)
Suppose not, i.e., hi(vi , v-i)  hi(v’i , v-i)
Case 1. s(vi , v-i) = s(v’i , v-i). If f is truthfully implementable in dominant strategies, we have
that vi(s(vi , v-i)) + mi(vi , v-i)  vi(s(v’i , v-i)) + mi(v’i , v-i) and
that v’i(s(v’i , v-i)) + mi(v’i , v-i)  v’i(s(vi , v-i)) + mi(vi , v-i)
Since s(vi , v-i) = s(v’i , v-i), these inequalities imply hi(vi , v-i) = hi(v’i , v-i). Contradiction

23. Uniqueness of Groves mechanism…
PROOF CONTINUES…
Case 2. s(vi , v-i)  s(v’i , v-i). Suppose wlog that hi(vi , v-i) > hi(v’i , v-i)
Consider an agent with the following valuation function:
Let v’’i(x) = - ji vj(s(vi , v-i)) if x = s(vi , v-i)
Let v’’i(x) = - ji vj(s(v’i , v-i)) +  if x = s(v’i , v-i)
Let v’’i(x) = - otherwise
We will show that v’’i will prefer to report vi for small 
Truth-telling being dominant requires
v’’i(s(v’’i , v-i)) + mi(v’’i , v-i) ≥ v’’i(s(vi , v-i)) + mi(vi , v-i)
s(v’’i , v-i) = s(v’i , v-i) since setting x = s(v’i , v-i) maximizes v’’i(x) + ji vj(x)
(This choice gives welfare , s(vi , v-i) gives 0, and other choices give - )
So, v’’i(s(v’i , v-i)) + mi(v’’i , v-i) ≥ v’’i(s(vi , v-i)) + mi(vi , v-i)
From which we get by substitution:
- ji vj(s(v’i , v-i)) +  + mi(v’’i , v-i) ≥ - ji vj(s(vi , v-i)) + mi(vi , v-i) 
- ji vj(s(v’i , v-i)) +  + ji vj(s(v’’i , v-i)) + hi(v’’i, v-i) ≥ -ji vj(s(vi , v-i)) +ji vj(s(vi , v-i)) + hi(vi, v-i)
  + hi(v’’i , v-i) ≥ hi(vi , v-i)
Because s(v’’i , v-i) = s(v’i , v-i), by the logic of Case 1, hi(v’’i , v-i) = hi(v’i , v-i)
This gives  + hi(v’i , v-i) ≥ hi(vi , v-i)
But by hypothesis we have hi(vi , v-i) > hi(v’i , v-i), so there is a contradiction for small  ■
Caveat to the theorem: Other mechanisms can work too if v is not unrestricted (or if the objective is not social welfare maximization)

24. Clarke tax “pivotal” mechanism
Special case of Groves mechanism: hi(v-i) = - ji vj(s(v-i))
So, agent’s payment mi = ji vj(s(v)) - ji vj(s(v-i))  0 is a tax
Intuition: Agent internalizes the negative externality he imposes on others by affecting the outcome
Agent pays nothing if he does not change (“pivot”) the outcome
Example 1: Second-price auction
Example 2: k=1, x1=“joint pool built” or “not”, mi = $
E.g. equal sharing of construction cost: -c / |A|, so vi(x1) = wi(x1) - c / |A|
So, ui = vi (x1) + mi
No pool
Pool $0 ui =5
=10 u
i =10
General preferences
Quasilinear preferences

25. Clarke tax mechanism… Pros Cons Social welfare maximizing outcome
Truth-telling is a dominant strategy
Ex post individually rational (i.e., even in hindsight each agent is no worse off by having participated)
Not all Groves mechanisms have this property, but Clarke tax does
Feasible in that it does not need a benefactor (i mi  0)
Cons
Budget balance not maintained (in pool example, generally i mi < 0)
Have to burn the excess money that is collected
Thrm. [Green & Laffont 1979]. Let the agents have quasilinear preferences ui(x, m) = mi + vi(x) where vi(x) are arbitrary functions. No social choice function that is (ex post) welfare maximizing (taking into account money burning as a loss) is implementable in dominant strategies
See also recent work on redistribution mechanisms by, e.g., Conitzer, Cavallo, …
If there is some party that has no private information to reveal and no preferences over x, welfare maximization and budget balance can be obtained by having that party’s payment be m0 = - i=1.. mi
E.g. auctioneer could be agent 0
Vulnerable to collusion
Even by coalitions of just 2 agents