Payoffs in Location Games

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

3
An artists rendering of Manhattan
Sams Starcups
Trudys Tazzo
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

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

6
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)

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

8
Some more notation

• PM(S) minL,F minn minT p(M,L,F)(S)
• PM(S) is the worst case performance of strategy
  S on any metric space in M
• PM maxS PM(S)
• PM(1) defined analogously when n1

9
Our Results

• PR d(1) 1/(d1)
• ½ 1/(d1) PR d 1/(d1)

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

11
The 1-D case Beaches Icecream

• No subsequent move of T can get ½
• Recall Ts strategy to obtain ½ repicate Ss
  moves
• S can use the same strategy for moves si1
• s1 ½ si ti-1

12
The 1-D case Beaches Icecream

• p(tn) ½
• p(t1,,tn-1) p(s2,,sn)
• ) p(S) p(s2,,sn) ¼

13
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

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

15
PR 2(1) 1/3
Condorcét voting paradox
D1 L1 L3 L2 D2 L2 L1 L3 D3 L3 L2
L1
The vote is inconclusive
16
PR 2(1) 1/3

• Our goal
• 9 a location s such that 8 t, p(s,t) 1/3
• Outline
• Construct a digraph on locations
• G contains edge u!v iff p(u,v)
• Show that G contains no cycles
• ) G has a sink s

17
PR 2(1) 1/3

• Each edge defines a half-space containing at
  least 2/3 of the demand
• A cycle defines an intersection of half-spaces

18
PR 2(1) 1/3
All triplets of half spaces must intersect!
Contradiction!!
19
PR 2(1) 1/3

• Hellys 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 Å1in Ci ¹

20
PR 2(1) 1/3

• Let u1,,uk be a cycle in G
• Then, d(P,ui1) d(P,ui)
• because P is in the half-space defined by the
  edge ui!ui1
• ) d(P,ui) d(P,uj) 8 i,j

21
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!!
22
The d-dimensional case

• Results on R 2 extend nicely to R d
• Thm 3 PR d(1) 1/(d1)
• Thm 4 1/2(d1) PR d 1/(d1)

23
Condorcét instance in d-dimensions

• As before we should have
• Di Li Li1 Li2 Li-1
• Embedding in R d1
• Li ( 0 , , 0 , 1 , 0 , , 0)
• Di (d-i,d-i1,,1-e,2, 3 , )
• Project all points down to the d dimensional
  plane containing L1,,Ld1 relative distances
  between Li and Dj are preserved

24
Lower bound in d-dimensions
(Skip)

• As before, define a directed graph on locations
  with each half-space containing d/(d1) demand
• Every set of d half-spaces must intersect
• By Hellys Theorem all half-spaces must have a
  non empty intersection. Assume WLOG that the
  origin lies in this intersection.

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 d1
  half-spaces
• Arrive at a contradiction just as before.

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, pni 0 8 i åi ni 0
• 9 a collection of d1 vectors ni such that
  åbini 0 with bi 0
• Then, å pbini 0.
• But pni 0 8 i, so, pni 0 for the d1
  vectors.
• Thus every point in the intersection of these
  half-spaces is equi-distant from all vertices in
  the cycle.

27
Lower bound gritty detail

• For any n vectors ni in d dimensions with some
  positive linear combination summing to zero, 9 a
  positive linear combination of some d1 of them
• Take some linear combination and eliminate the
  most negative term iterate

28
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.

29
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.

30
Questions?