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### Dueling Algorithms

Nicole Immorlica, Northwestern University

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

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 Contexts

Ranking games [BFHS’08]:

Players choose strategies to achieve particular payoff rank among peers.

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

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 Representations

Optimization duels [this work]:

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

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 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.

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

Search for “pretty shape”.

See which search engine ranks my favorite shape higher.

Thereafter, use that one.

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

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?

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 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

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

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

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 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 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 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 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

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 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 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

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

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

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-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

- 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

- 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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