Computational Issues in Game Theory Lecture 3: Other topics

1
Computational Issues in
Game Theory Lecture 3
Other topics

• Edith Elkind
• Intelligence, Agents, Multimedia group (IAM)
• School of Electronics and CS
• U. of Southampton

2
Plan

• Selfish routing
• Voting and voting manipulation
• Coalitional games

3
Selfish routing model

• Intuition traffic from A to B
• individual drivers make decisions
• no singe car has a perceivable effect
• A directed graph G (V, E)
• Source-sink pairs (si, ti) for i1, , k
• rate ri 0 of traffic between si and ti
• For each edge e, a latency function le(.)
• assume le(.) 0, monotone, differentiable

r1 1
4
Non-atomic Players

• Routing game infinitely many players of
  infitezimal size (particles of flow)
• individual player cannot affect the payoff for
  others, but can affect its own payoff
• can be viewed as a limit of an
  atomic game as n ? infinity

5
Selfish Flows
r 1
Goal minimize average travel time

• upper edge x
• lower edge 1-x
• average travel time 1/21/2 11/2 3/4
• But traffic on the lower edge is envious!

xx1(1-x) 2x-1 x ½
Envy-free (selfish) flow average travel time 1
6
Selfish Agents

• Total latency is a measure of social welfare.
• Agents are selfish
• do not care about social welfare
• want to minimize personal latency
• (Wardrop)-Nash equilibrium no one wants
  to change strategy, i.e., switch paths
• equivalently all flow paths from si to ti
  have the same travel
  time.
• unique WNE in pure strategies
• Cost of anarchy Worst Nash/Opt ?
• cost of stability Best Nash/Opt

7
Cost of Anarchy General Latencies

• Previous examples cost of anarchy 4/3
• Corollary for any C gt 0, the cost of anarchy is
  C.
• What can we do now?

Opt xxd1(1-x) (d1)xd1-1 xOPT
1/(d1)1/(d1) 1 o(1) C(f) 1/(d1) o(1)
? 0
Nash all traffic chooses upper path, C(f) 1
8
The price of anarchy bounds

• Bicriteria bound
  Theorem if f is a NE flow for (G, r,
  l) and f is feasible for (G,
  2r, l), then C(f) C(f).
• Linear latencies
  Theorem if f is a NE flow for (G, r, l),
  f is an optimal flow for
  (G, r, l), and all
  latencies are linear, then
  C(f) 4/3 C(f).

9
Braesss Paradox
.5
Opt Nash, Latency 1.5
.5
What if we build a new road?
.5
0
Latency 1.75
Latency 1.5
Latency 2
.5
10
Severity of Braesss paradox

• Let f be a NE flow for (V, E),
  gE is a NE flow for (V, E), E ? E
• what is maxEC(f)/C(gE)?
• n/2
• k (if you can only delete
  k edges)
• can we detect Braesss paradox?
• i.e., is there an E ? E s.t. C(gE) lt C(f)?
• no NP-hard

11
Taxes can help!
C

• Braesss paradox
  assign a tax of 1
  to CD
• the flow is optimal now!
• can we ensure optimal flow
  by taxing the traffic?
• yes marginal cost pricing (MCP)
• intuition each user pays for the extra delay
  they cause
• if users minimize latencytax, MCP induces
  optimal flow
• maybe at the expense of large taxes
• alternative goal minimize total disutility

x
0
A
B
0
0
x
D
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Voting

• n voters
• m candidates c1, , cm
• each voter has a preference over candidates
• total ordering ci gt cj gt
• voting rule
  voters preferences ? candidates
• or preferences ? total orders of candidates

13
Voting rules examples

• Plurality each candidate gets 1 point from each
  voter that ranks him first
• Borda each candidate gets
• m-1 point from each voter that ranks him 1st
• m-2 points from each voter that ranks him 2nd
• Veto each candidate gets 1 point from each voter
  that does not rank him last
• Scoring rules each candidate gets
• ai points from each voter that ranks him ith

14
Voting rules binary cup
Do most voters prefer A to B?
R1
F
R2
C
F
R3
C
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Voting rules more examples

• Copeland
• is score j majority prefers i to j
• candidate with the highest score wins
• Maximin
• is score min j v v prefers i to j
• candidate with the highest score wins
• STV
• if someone has more than n/2 votes, he wins
• else, find a candidate with the least number of
  1st place votes and cross him out
  from all ballots
• repeat until there is a majority winner

16
Voting rules criteria

• Pareto-optimality
• if everyone prefers i to j, j does not win
• not true for Binary Cup
• Condorset-consistency
• if i is such that for each j
  majority of voters
  prefers i to j, then i wins
• not true for Borda
• Monotonicity
• if we move the winning candidate up in one of the
  votes, he still wins
• not true for STV

17
Arrow theorem

• Let F output total orderings of candidates
• Independence of irrelevant alternatives (IIA)
• for any
• two profiles R(R1, , Rm) and S(S1, , Sm)
• pair of candidates a, b
• such that
• i prefers a to b in R iff i prefers a to b in S
  (for each i)
• F(R) and F(S) order a and b in the same way
• Theorem there is no F for m 3 candidates that
  is
• non-dictatorial
• Pareto-optimal
• satisfies IIA

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Manipulation

• Can in be in the voters best interest to
  misrepresent his preferences?
• 50 voters AgtBgtC
• 50 voters CgtAgtB
• 20 voters BgtAgtC
• voting rule plurality (draws resolved by a coin
  toss)
• any voter with preference BgtAgtC
  is better off
  voting AgtBgtC
• in this scenario, STV is truthful
• but there are other examples

19
Gibbard-Satterthwaite theorem

• For any voting rule for m3 candidates that is
• non-dictatorial
• symmetric wrt candidates
• there is a set of preference profiles such that
• one of the voters is better off voting
  non-truthfully
• Extends to randomized voting rules
• nonmanipulable rules are all of the form
• w.p. p, select a random voter and elect his top
  candidate
• w.p. 1-p, select a random pair of candidates,
  and elect
  the better of the two

20
Escaping GS theorem computational barriers

• Plurality is easy to manipulate
• rank your favorite electable candidate first
• STV less so
• STV is NP-hard to manipulate
  Bartholdi, Orlin91
• General methods for constructing
  manipulation-resistant voting schemes Conitzer,
  Sandholm03, Elkind, Lipmaa05

21
Escaping GS theorem structured
preferences
c1
c4
c3
c2
c5
c6
c7
cv

• Candidates are ordered from left to right
• Each voter v has a favorite candidate cv
• If ci is between cv and cj, then ci gtv cj
• v may prefer c7 to c2
• Truthful voting rule
• order voters by their favorite candidates
• winner favorite candidate of the median voter

22
Coalitional games

• Intuition players can cooperate to achieve goals
  and then share the payoff
• Payoff function coalitions ? utilities
• transferable utility each coalition has a value
• it is assumed that the players can share the
  value
• example team of workers building a house
• non-transferable utility for each coalition,
  there is a value for each player
• example trade agreement between countries

23
Coalitional games examples

• Weighted voting games
• n players with weights w1, , wn, quota T
• w(C) Si?Cwi
• v(C)1 (C wins) if w(C) T, v(C)0 otherwise
• Network flow games
• network G(V, E), with source s and sink t, edge
  capacities c1, , cE
• players are edges
• v(C) is the max s-t flow that C can carry

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Coalitional games special classes

• Simple games value of each C is 0 or 1
• weighted voting games
• variant of network flow games (thresholded)
• Monotone games if S ? T, then v(S) v(T)
• weighted voting games with non-negative weights
  (but not general WV games)
• Superadditive games
  if SnTØ, then
  v(S U T) v(S)v(T)
• network flow games
• usually not the case for simple games
• can assume that the grand coalition will form

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Dummy players

• a is a dummy if for any C, v(CUa)v(C)
• Weighted voting games e.g.,
• a has weight 1
• all other players have even weight
• quota is even
• Network flow games
• no loop-free s-t path containing a
• NP-hard to detect in many games
• e.g., WV games, NF games

a
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How to distribute payoffs fairness

• Shapley value
  agents expected contribution
• Intuition agents join coalition
  in random order
• how does the value change when i joins?
• Formally
• p a permutation of n players
• Pred(i, p) j p( j ) lt p( i )
  
• fi(v) 1/n! Sp v(Pred(i, p)U i ) -
  v(Pred(i, p)

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Shapley value
axiomatic characterization

• efficiency Si fi(v) v(grand coalition)
• dummy property dummy players receive 0
• symmetry if for any C s.t. i, j not in C
  we have v(C U i ) v(C U j ),
  then fi(v)
  fj(v)
• additivity for two coalitional games v, w,
  define vw by (vw)(C) v(C)w(C).
  We
  have fi(vw) fi(v)fi(w)
• Shapley value is the only value distribution
  scheme that satisfies 1 - 4 !

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Computing Shapley value

• Weighted voting games
• computing Shapley value is NP-hard
• hard to decide if i is a dummy (i.e., fi 0)
• unless PNP, there is no algorithm for computing
  fi in time poly(n, log max wi)
• poly(n, max wi) algorithms exist
• for w1, , max wi, count coalitions of weight
  wi
• dynamic programming
• Network flow games
• computing Shapley value is NP-hard
• hard to decide if an edge is a dummy

29
How to distribute payoffs
stability

• p (p1, , pn) payoff vector
• core the set of payoff vectors such that noone
  wants to deviate
• p is in the core if for any J we have p(J) v(J)
• Core can be empty
• Lemma in simple games,
  the core is empty ltgt
  no player belongs
  to
  all winning coalitions

pj gt 0
J
p(J) lt 1
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Relaxing the Notion of the Core

• e- core p is in the e- core iff for any J
  p(J) v(J) - e
• each coalition C gets at least v(C) - e
• nonempty for large enough e
• least core smallest non-empty e-core
• if least core e - core
• there is a p s.t. p(J) v(J) - e for any J
• for any e lt e there is no p
  s.t. p(J)
  v(J) - e for any J

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Coalition structures motivation

• Definitions of Shapley value and core
  distribute the value of the grand
  coalition
• makes sense for superadditive games
• less so for simple games
• Can several coalitions form simultaneously?
• weighted voting games
• suppose C1 and C2 are disjoint, w(C1) gt T, w(C2)
  gt T
• merging C1 and C2 is bad for social welfare
• Sometimes, grand coalition cannot form
• legal restrictions, cant negotiate

32
Coalition structures definition

• Coalition structure CS C1, , Ck
  a partition of all agents
  into coalitions
• each agent belongs to some Ci
• Payoffs are distributed within coalitions
• p has to satisfy p(Ci)v(Ci) for i 1, , k
• CS-core (CS, p) is in CS-core (i.e.,stable) if
  no group of agents wants to deviate
• for every S, p(S) v(S)

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CS-core vs. core

• Weighted voting game
• 5 players a, b, c, d, e
• weights 2, 2, 2, 3, 3 T 6
• 2 disjoint winning coalitions a, b, c, d, e
• core is empty
• ½ -core is non-empty (1/6, 1/6, 1/6, 1/4, 1/4)
• CS-core is non-empty
  CSa, b, c, d, e, p(1/3, 1/3,
  1/3, 1/2, 1/2)