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Payoffs in Location Games

Payoffs in Location Games. Shuchi Chawla 1/22/2003 joint work with Amitabh Sinha, Uday Rajan & R. Ravi. Caffeine wars in Manhattan. S am owns Starcups ; T rudy owns Tazzo Every month both chains open a new outlet – S am chooses a location first, T rudy follows

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Payoffs in Location Games

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  1. Payoffs in Location Games Shuchi Chawla 1/22/2003 joint work with Amitabh Sinha, Uday Rajan & R. Ravi

  2. Caffeine wars in Manhattan • Sam owns Starcups; Trudy owns Tazzo • Every month both chains open a new outlet – Sam chooses a location first, Trudy follows • Indifferent customers go to the nearest coffee shop • At the end of n months, how much market share can Sam have? • Trudy knows n, Sam doesnt Shuchi Chawla, Carnegie Mellon Univ

  3. T T T $$ $$ $$ An artist’s rendering of Manhattan Sam’s Starcups Trudy’s Tazzo Shuchi Chawla, Carnegie Mellon Univ

  4. Why bother? • Product Placement • many features to choose from – high dimension • high cost of recall – cannot modify earlier decisions • Service Location • cannot move service once located Shuchi Chawla, Carnegie Mellon Univ

  5. Some history… • The problem was first introduced by Harold Hotelling in 1929 • Acquired the name “Hotelling Game” • Originally studied on the line with n players moving simultaneously • Extensions to price selection Shuchi Chawla, Carnegie Mellon Univ

  6. p(M,L,F)(S) = sd(v,S)<d(v,T)dF(v) + ½sd(v,S)=d(v,T)dF(v) Formally… • Given:(M,L,F) – Metric space, Location set, Distribution of demands • At step i, S first picks si2 L. Then T picks ti2 L • si = si(s1,…,si-1,t1,…,ti-1) ; ti = ti(s1,…,si,t1,…,ti-1) • S is an online player: does not know n • Payoff for S at the end of n moves is: • p(M,L,F)(T) = 1 - p(M,L,F)(S) Shuchi Chawla, Carnegie Mellon Univ

  7. The second mover advantage • Note that if 8 i, ti = si p(M,L,F)(S) = p(M,L,F)(T) = ½ • T can always guarantee a payoff of ½ • Can S do the same? • We will show that S cannot guarantee ½ but at least some constant fraction depending on M Shuchi Chawla, Carnegie Mellon Univ

  8. Some more notation • PM(S)= minL,F minn minTp(M,L,F)(S) • PM(S) is the worst case performance of strategy S on any metric space in M • PM= maxSPM(S) • PM(1) – defined analogously when n=1 Shuchi Chawla, Carnegie Mellon Univ

  9. Our Results PR d(1) = 1/(d+1) ½ 1/(d+1) ·PR d· 1/(d+1) Shuchi Chawla, Carnegie Mellon Univ

  10. The 1-D case: Beaches & Icecream • Assume a uniform demand distribution for simplicity • S moves at ½ , no move of T can get more than ½ )PR (1) = ½ Shuchi Chawla, Carnegie Mellon Univ

  11. The 1-D case: Beaches & Icecream • No subsequent move of T can get > ½ • Recall T’s strategy to obtain ½ : repicate S’s moves • S can use the same strategy for moves si>1 s1 = ½ ; si = ti-1 Shuchi Chawla, Carnegie Mellon Univ

  12. The 1-D case: Beaches & Icecream • p(tn) · ½ • p(t1,…,tn-1) = p(s2,…,sn) )p(S) ¸p(s2,…,sn) ¸ ¼ Shuchi Chawla, Carnegie Mellon Univ

  13. “MEDIAN” “REPLICATE” Median and Replicate • Given a 1-move strategy with payoff r obtain an n-move strategy with payoff r/2 • Use 1-move strategy for the first move, Replicate all other moves of player 2 • Last move of player 2 gets at most 1-r, the rest get at most half of the remaining : r/2 Shuchi Chawla, Carnegie Mellon Univ

  14. Locn Game in the Euclidean plane • Thm 1: PR 2(1)= 1/3 • Thm 2: 1/6 ·PR 2· 1/3 • Proof of Thm 2: Use Median and Replicate with Thm 1 Shuchi Chawla, Carnegie Mellon Univ

  15. L1 D1 T D2 L3 $$ L2 D3 S gets only 1/3 of the demand PR 2(1) · 1/3 Condorcét voting paradox D1 : L1 > L3 > L2 D2 : L2 > L1 > L3 D3 : L3 > L2 > L1 The vote is inconclusive Shuchi Chawla, Carnegie Mellon Univ

  16. PR 2(1) ¸ 1/3 • Our goal: 9 a location s such that 8t, p(s,t) ¸ 1/3 • Outline: Construct a digraph on locations G contains edge u!v iff p(u,v) < 1/3 Show that G contains no cycles )G has a sink s Shuchi Chawla, Carnegie Mellon Univ

  17. Each edge defines a half-space containing at least 2/3 of the demand A cycle defines an intersection of half-spaces > 2/3 demand PR 2(1) ¸ 1/3 Shuchi Chawla, Carnegie Mellon Univ

  18. If not: < 1/3 < 1/3 < 1/3 PR 2(1) ¸ 1/3 All triplets of half spaces must intersect! Contradiction!! Shuchi Chawla, Carnegie Mellon Univ

  19. ) all half-spaces defined by the graph G contain a common point P PR 2(1) ¸ 1/3 • Helly’s Theorem Given a collection {C1,C2,…,Cn} of convex sets in Euclidean space : If every triplet of the sets has a non empty intersection, then Å1·i·nCi¹; Shuchi Chawla, Carnegie Mellon Univ

  20. P Pis the intersection of hyperplanes bisecting the edges PR 2(1) ¸ 1/3 • Let u1,…,uk be a cycle in G • Then,d(P,ui+1) · d(P,ui) because P is in the half-space defined by the edge ui!ui+1 )d(P,ui) = d(P,uj) 8 i,j Shuchi Chawla, Carnegie Mellon Univ

  21. P >1/3 > 2/3 - a/2 PR 2(1) ¸ 1/3 Let demand at P be a. Then each half-space has a total demand of at least 2/3 + a/2 Contradiction!! Shuchi Chawla, Carnegie Mellon Univ

  22. The d-dimensional case • Results on R 2 extend nicely to R d: Thm 3 : PR d(1) = 1/(d+1) Thm 4 : 1/2(d+1) ·PR d· 1/(d+1) Shuchi Chawla, Carnegie Mellon Univ

  23. Condorcét instance in d-dimensions • As before we should have Di: Li > Li+1 > Li+2 > … > Li-1 • Embedding in Rd+1 : • Li´ ( 0 , … , 0 , 1 , 0 , … , 0) • Di´ (d-i,d-i+1,…,1-e,2, 3 , … ) • Project all points down to the d dimensional plane containing {L1,…,Ld+1} – relative distances between Li and Dj are preserved Shuchi Chawla, Carnegie Mellon Univ

  24. Lower bound in d-dimensions (Skip) • As before, define a directed graph on locations with each half-space containing > d/(d+1) demand • Every set of d half-spaces must intersect • By Helly’s Theorem all half-spaces must have a non empty intersection. Assume WLOG that the origin lies in this intersection. Shuchi Chawla, Carnegie Mellon Univ

  25. Lower bound in d-dimensions • Assume for the sake of contradiction that a cycle exists. • Each point in the intersection is equi-distant from all vertices in the cycle • We want this to hold for at most some d+1 half-spaces • Arrive at a contradiction just as before. Shuchi Chawla, Carnegie Mellon Univ

  26. Lower bound in d-dimensions • Let ni be a vector representing the i-th edge in the cycle; Let p represent some point in the intersection • Then, p¢ni¸ 0 8 i; åini = 0 • 9 a collection of · d+1 vectors ni such that åbini = 0 with bi¸ 0 • Then, å p¢bini = 0. • But p¢ni¸ 0 8 i, so, p¢ni = 0 for the d+1 vectors. • Thus every point in the intersection of these half-spaces is equi-distant from all vertices in the cycle. Shuchi Chawla, Carnegie Mellon Univ

  27. Concluding Remarks • Results hold even when demands lie in some high dimensional space • We can obtain tighter results in the line when n is bounded. Shuchi Chawla, Carnegie Mellon Univ

  28. Open Problems • Closing the factor-of-2 gap for P • Convergence with n • If S knows a lower/upper bound on n, is there a better strategy? • Can he do better as n gets larger – we believe so • Brand loyalty • What about demand in the intermediate steps? • a fraction of demand at every time step becomes loyal to the already opened locations. The rest carries on to the next step. Shuchi Chawla, Carnegie Mellon Univ

  29. Questions? Shuchi Chawla, Carnegie Mellon Univ

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