Introduction The Model Characterization of NE Quality of NE Conclusion Voronoi Games on Cycle Graphs Marios Mavronicolas Burkhard Monien Vicky G. Papadopoulou Florian Schoppmann Department of Computer Science International Graduate School Dynamic Intelligent Systems University of Paderborn August 26, 2008 Intern. Grad. School, University of Paderborn Florian Schoppmann 1 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Competitive Location Two ice cream vendors on the beach: ◮ Customers typically buy at nearest vendor ◮ People are spread evenly ◮ What are the optimal positions? ◮ Vendors have an incentive to move towards each other Water ◮ Hotelling’s law (1929): It is Beach rational for producers to 100m Street 0m 50m make products as similar as possible Intern. Grad. School, University of Paderborn Florian Schoppmann 2 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Some non-scientific references to Hotelling’s law... Los Angeles Times, June 8, 2008 (editorial): “Obama and McCain, the same?” [...] With a Republican president experiencing some of the worst approval ratings ever, it’s no shock that the party opted for an unusually centrist candidate. Yet Obama, too, represents a break from Democratic orthodoxy and is reaching out to the middle . [...] Some might complain that this means voters will have little to choose between in November. We say: Welcome to the middle , candidates. [...] Intern. Grad. School, University of Paderborn Florian Schoppmann 3 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Motivation and Framework Hotelling’s law: ◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions Classification by Eiselt et al. (1993) with over 100 references: 1. underlying metric measurable space 2. number of players 3. pricing policy (if any) 4. the equilibrium concept 5. customers’ behavior They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Motivation and Framework Hotelling’s law: ◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions Classification by Eiselt et al. (1993) with over 100 references: 1. underlying metric measurable space 2. number of players 3. pricing policy (if any) 4. the equilibrium concept 5. customers’ behavior They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Motivation and Framework Hotelling’s law: ◮ “principle of minimum differentiation” (Boulding, 1966) ◮ Very sensitive to original assumptions Classification by Eiselt et al. (1993) with over 100 references: 1. underlying metric measurable space 2. number of players 3. pricing policy (if any) 4. the equilibrium concept 5. customers’ behavior They call CL one of the “truly interdisciplinary fields of study” Motivation for computer science: Competitive service providers in discrete networks Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion The Model Definition A Voronoi game (on a connected undirected graph) is specified by: ◮ Graph G = ( V , E ) ◮ number of players k ≤ | V | The strategic game is then: ◮ Strategy set of all players is V , set of profiles is S := V k ◮ Utilities: ◮ Define F v := S → 2 [ k ] , F v ( x ) := arg min i ∈ [ k ] dist ( v , s i ) mapping each node to the set of nearest players ◮ Utility of a player i in profile s is: 1 � u i ( s ) := | F v ( s ) | v ∈ V : i ∈ F v ( s ) Intern. Grad. School, University of Paderborn Florian Schoppmann 5 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Nash Equilibrium Situation (= strategy profile) where no player can improve: 3 3 ◮ u i ( s − i , s ′ i ) ≤ u i ( s ) holds for all players i and all alternative strategies s ′ i Intern. Grad. School, University of Paderborn Florian Schoppmann 6 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Social Cost and Example Social cost is total transport cost: Cycle graph, 6 players: � SC ( s ) := i ∈ [ k ] { dist ( v , s i ) } min 4 3 v ∈ V 1 4 Optimum is OPT := min s ∈ S { SC ( s ) } 1 3 2 Loss due to selfish behavior captured by prices of anarchy and stability: SC ( s ) SC = 3 , OPT = 2 PoA := max OPT s is NE SC ( s ) PoS := min OPT s is NE ( 0 0 := 1 and x 0 := ∞ here) Intern. Grad. School, University of Paderborn Florian Schoppmann 7 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Related Work (1/2) Eaton and Lipsey (1975): Voronoi games on a continuous circle s 1 s 1 ,s 3 s 3 s 3 0 0 0 s 2 s 2 s 2 s 1 ◮ A profile is a NE iff “no firm’s whole market is smaller than any other firm’s half market” ◮ In a cost-maximizing NE: # players k is even, all players are paired, pairs are equidistantly located ◮ In an optimum: all players equidistantly located = ⇒ PoA = 2 and PoS = 1 Intern. Grad. School, University of Paderborn Florian Schoppmann 8 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Related Work (2/2) Only one work on discrete Voronoi games (Dürr and Thang, ESA 2007): ◮ A graph without NE, even for k = 2 ◮ Best response dynamics do not converge in general, not even on cycle graphs ◮ Deciding existence of NE is NP-hard ◮ Bound on ratio of social cost in worst/best NE Voronoi games on graphs also similar to (but different than) competitive facility location games (Vetta, FOCS 2002) Intern. Grad. School, University of Paderborn Florian Schoppmann 9 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Cycle Graphs C ( n , k ) Major difference to circle: ◮ Points equidistant to more than θ 0 d 0 = 2 one player have non-zero measure θ 4 Convenient representation of profile s : θ 1 ◮ θ 0 , . . . , θ ℓ − 1 are used nodes (in θ 3 counterclockwise orientation) ◮ d i = dist ( θ i , θ i + 1 ) θ 2 ◮ c i is the number of players on θ i ◮ Up to rotation/renumbering, ℓ , ( d i ) i ∈ Z ℓ , and ( c i ) i ∈ Z ℓ suffice Intern. Grad. School, University of Paderborn Florian Schoppmann 10 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Characterization Theorem A profile with minimum utility γ is a NE if and only if ∀ i ∈ Z ℓ : 1. c i ≤ 2 2. d i ≤ 2 γ 3. c i � = c i + 1 = ⇒ ⌊ 2 γ ⌋ odd 4. c i = 1 , d i − 1 = d i = 2 γ = ⇒ 2 γ odd 5. c i = c i + 1 = 1 , d i − 1 + d i = d i + 1 = 2 γ = ⇒ 2 γ odd c i = c i − 1 = 1 , d i − 1 = d i + d i + 1 = 2 γ = ⇒ 2 γ odd Sketch of “ ⇒ ”-proof (only for γ = 1). … … Intern. Grad. School, University of Paderborn Florian Schoppmann 11 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Characterization Theorem A profile with minimum utility γ is a NE if and only if ∀ i ∈ Z ℓ : 1. c i ≤ 2 2. d i ≤ 2 γ 3. c i � = c i + 1 = ⇒ ⌊ 2 γ ⌋ odd 4. c i = 1 , d i − 1 = d i = 2 γ = ⇒ 2 γ odd 5. c i = c i + 1 = 1 , d i − 1 + d i = d i + 1 = 2 γ = ⇒ 2 γ odd c i = c i − 1 = 1 , d i − 1 = d i + d i + 1 = 2 γ = ⇒ 2 γ odd Sketch of “ ⇒ ”-proof (only for γ = 1). … … Intern. Grad. School, University of Paderborn Florian Schoppmann 11 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Characterization Theorem A profile with minimum utility γ is a NE if and only if ∀ i ∈ Z ℓ : 1. c i ≤ 2 2. d i ≤ 2 γ 3. c i � = c i + 1 = ⇒ ⌊ 2 γ ⌋ odd 4. c i = 1 , d i − 1 = d i = 2 γ = ⇒ 2 γ odd 5. c i = c i + 1 = 1 , d i − 1 + d i = d i + 1 = 2 γ = ⇒ 2 γ odd c i = c i − 1 = 1 , d i − 1 = d i + d i + 1 = 2 γ = ⇒ 2 γ odd Sketch of “ ⇒ ”-proof (only for γ = 1). … … Intern. Grad. School, University of Paderborn Florian Schoppmann 11 / 22 ·
Introduction The Model Characterization of NE Quality of NE Conclusion Characterization Theorem A profile with minimum utility γ is a NE if and only if ∀ i ∈ Z ℓ : 1. c i ≤ 2 2. d i ≤ 2 γ 3. c i � = c i + 1 = ⇒ ⌊ 2 γ ⌋ odd 4. c i = 1 , d i − 1 = d i = 2 γ = ⇒ 2 γ odd 5. c i = c i + 1 = 1 , d i − 1 + d i = d i + 1 = 2 γ = ⇒ 2 γ odd c i = c i − 1 = 1 , d i − 1 = d i + d i + 1 = 2 γ = ⇒ 2 γ odd Sketch of “ ⇒ ”-proof (only for γ = 1). … … Intern. Grad. School, University of Paderborn Florian Schoppmann 11 / 22 ·
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