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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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Color test color test color test color test

COLOR TESTCOLOR TESTCOLOR TESTCOLOR TEST


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.


Color test color test color test color test

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.


Color test color test color test color test

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 response1

Compression Best-Response

Depth:

1

2

3

4


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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