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COLOR TEST COLOR TEST COLOR TEST COLOR TEST. Dueling Algorithms. Nicole Immorlica , Northwestern University with A. Tauman Kalai , B. Lucier , A. Moitra , A. Postlewaite , and M. Tennenholtz. Social Contexts. Normal-form games :

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dueling algorithms

Dueling Algorithms

Nicole Immorlica, Northwestern University

with A. TaumanKalai, B. Lucier, A. Moitra, A. Postlewaite, and M. Tennenholtz

social contexts
Social Contexts

Normal-form games:

Players choose strategies to maximize expected von Neumann-Morgenstern utility.

Social context games [AKT’08]:

Players choose strategies to achieve particular social status among peers.

social contexts1
Social Contexts

Ranking games [BFHS’08]:

Players choose strategies to achieve particular payoff rank among peers.

two player ranking games
Two-Player Ranking Games

Bob

G

Alice and Bob play game:

Alice

1

Alice beats Bob in G

½

Alice ties Bob in G

RG payoff of Alice:

0

Alice loses to Bob in G

implicit representations
Implicit Representations

Succinct games [FIKU’08]:

Payoff matrix represented by boolean circuit. NE hard to solve or approximate.

Blotto games [B’21, GW’50, R’06, H’08]:

Distribute armies to battlefields.

implicit representations1
Implicit Representations

Optimization duels [this work]:

Underlying game is optimization problem. Goal is to optimize better than opponent.

ranking duel
Ranking Duel

A search engine is an algorithm that inputs

  • set Ω = {1, 2, …, n} of items
  • probabilities p1 + … + pn = 1 of each

and outputs a permutation π of Ω.

Monopolist objective: minimize Ei~p[π(i)].

ranking duel1
Ranking Duel

Competitive objective: Let the expected score of a ranking π versus a ranking π’ be

Pri~p[ π(i) < π’(i) ] + (½) Pri~p[ π(i) = π’(i) ].

Then objective is to output a π that maximizes expected score given algorithm of opponent.

optimizing a search engine
Optimizing a Search Engine

?

User searches for object drawn according to known probability dist.

slide11

Greedy is optimal.

0.19

0.16

0.27

0.07

0.22

0.09

Search:

pretty shape

1.

(27%)

2.

(22%)

3.

(19%)

4.

(16%)

5.

(09%)

6.

(07%)

choosing a search engine
Choosing a Search Engine

Search for “pretty shape”.

See which search engine ranks my favorite shape higher.

Thereafter, use that one.

slide13

0.19

0.16

0.27

0.07

0.22

0.09

Search:

Search:

pretty shape

pretty shape

6.

1.

(27%)

(27%)

2.

1.

(22%)

(22%)

2.

3.

(19%)

(19%)

4.

3.

(16%)

(16%)

4.

5.

(09%)

(09%)

6.

5.

(07%)

(07%)

questions
Questions

Can we efficiently compute an equilibrium of a ranking duel?

How poorly does greedy perform in a competitive setting?

What consequences does the duel have for the searcher?

optimization problems as duels
Optimization Problems as Duels

Ranking

Binary Search

Routing

Finish

?

?

?

?

?

?

?

Start

Hiring

Compression

Parking

duel framework
Duel Framework

Finite feasible set X of strategies.

Prob. distribution p over states of nature Ω.

Objective cost c: Ω × X R.

Monopolist: choose x to minimize Eω~p[cω(x)].

duel framework1
Duel Framework

1 if cω(x) < cω(x’)

v(x,x’) = Eω~p

0 if cω(x) > cω(x’)

½ if cω(x) = cω(x’)

Players select strategies x, x’ from X.

Nature selects state ωfrom Ωaccording to p.

Payoffs v(x,x’), (1-v(x,x’)) are realized.

results computation
Results: Computation

An LP-based technique to compute exact equilibria,

A low-regret learning technique to compute approximate equilibria,

… and a demonstration of these techniques in our sample settings

computing exact equilibria
Computing Exact Equilibria

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.
  • Define constraints describing K=convex-hull(X).
  • Define payoff matrix M that computes values.
  • Maps points in K back to strategies in original setting.
bilinear duels
Bilinear Duels

If feasible strategies X are points in Rn, and payoff v(x, x’) is xtMx’ for some M in Rnxn, then

maxv,x v s.t.

xtMx’ ≥ v for all x’ in X

x is in K (=convex-hull(X))

Exponential, but equivalent poly-sized LP.

ranking duel2
Ranking Duel

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.

X = set of permutation matrices

(entry xij indicates item i placed in position j)

  • Define constraints describing K=convex-hull(X).

K = set of doubly stochastic matrices

(entry yij = prob. item i placed in position j)

ranking duel3
Ranking Duel

Formulate game as bilinear duel:

  • Design “rounding alg.” that maps points in K back to strategies in original setting.

Birkhoff–von Neumann Theorem: Can efficiently construct permutation basis for doubly stochastic matrix (e.g., via matching).

ranking duel4
Ranking Duel

Formulate game as bilinear duel:

  • Define payoff matrix M that computes values.

Ep,y,y’[v(x,x’)]

= ∑i p(i) ( ½ Pry,y’ [xi= x’i] + Pry,y’ [xi> x’i])

= ∑i p(i) (∑iyij ( ½ y’ij + ∑k>j y’ik))

which is bilinear in y,y’ and so can be written ytMy’.

ranking duel5
Ranking Duel

Result: Can reduce computation time to poly(n) versus poly(n!) with standard LP approach.

Technique also applies to hiring duel and binary search duel.

compression duel
Compression Duel

data

(each with prob. p(.))

Goal: smaller compression (i.e., lower depth in tree).

classical algorithm
Classical Algorithm

Huffman coding:

Repeatedly pair nodes with lowest probability.

compression duel1
Compression Duel

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.

X = subset of zero-one matrices*

(entry xij indicates item i placed at depth j)

  • Define constraints describing K=convex-hull(X).

K = subset of row-stochastic matrices*

(entry yij = prob. item i placed at depth j)

* Must correspond to depth profile of some binary tree!

compression duel2
Compression Duel

Formulate game as bilinear duel:

  • Define payoff matrix M that computes values.

Ep,y,y’[v(x,x’)] = ∑i p(i) (∑iyij ( ½ y’ij + ∑k>j y’ik))

which is bilinear in y,y’ and so can be written ytMy’.

compression duel3
Compression Duel

Bilinear Form:

maxv,x v s.t.

xtMx’ ≥ v for all x’ in X

x is in K (=convex-hull(X))

Problems:

1. How to round points in K back to a random binary tree with right depth profile?

2. How to succinctly express constraints describing K?

approximate minimax
Approximate Minimax

Defn. For any ε > 0, an approximate minimaxstrategy guarantees payoff not worse than best possible value minus ε.

Defn. For any ε > 0, an approximate best response has payoff not worse than payoff of best response minus ε.

best response oracle
Best-Response Oracle

Idea. Use approximate best-response oracle to get approximate minimax strategies.

1. Low-regret learning: if x1,…,xT and x’1,…,x’T have low regret, then ave. is approx minimax.

2. Follow expected leader: on round t+1, play best-response to x1,…,xt to get low-regret.

compression best response
Compression Best-Response

Multiple-choice Knapsack:

Given lists of items with values and weights, pick one from each list with max total value and total weight at most one.

compression best response2
Compression Best-Response

For j from 1..n, list of depth j:

v( ) = Pr[win at depth j | x’ ]

w( ) = 2-j

… Kraft inequality

(each with prob. p(.))

x’ in K

other duels
Other Duels
  • Hiring duel: constraints defining Euclidean subspace correspond to hiring probabilities.
  • Binary search duel: similar to hiring duel, but constraints defining Euclidean subspace more complex (must correspond to search trees).
  • Racing duel: seems computationally hard, even though single-player problem easy.
conclusion
Conclusion
  • Every optimization problem has a duel.
  • Classic solutions (and all deterministic algorithms) can usually be badly beaten.
  • Duel can be easier or harder to solve, and can lead to inefficiencies.

OPEN QUESTION: effect of duel on the solution to the optimization problem?

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