1. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 frank@micc.unimaas.nl One-Way Flow Nash Networks Frank Thuijsman Joint work with Jean Derks, Jeroen Kuipers, Martijn Tennekes

2. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 2 Outline The Model of One-Way Flow Networks An Existence Result A Counterexample

3. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 3 Outline The Model of One-Way Flow Networks An Existence Result A Counterexample

4. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 4 Outline The Model of One-Way Flow Networks An Existence A Counterexample

5. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 5 The Model of One-Way Flow Networks Network Formation Game (N,v,c)

6. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 6 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n}

7. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 7 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j

8. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 8 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j

9. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 9 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j Example of a network g 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

10. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 10 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j Example of a network g Agent 1 is connected to agents 3,4,5 and 6 and obtains profits v 13, v 14, v 15, v 16 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

11. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 11 Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j Example of a network g Agent 1 is connected to agents 3,4,5 and 6 and obtains profits v 13, v 14, v 15, v 16 Agent 1 is not connected to agent 2 The Model of One-Way Flow Networks 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

12. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 12 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j Example of a network g Agent 1 is connected to agents 3,4,5 and 6 and obtains profits v 13, v 14, v 15, v 16 Agent 1 is not connected to agent 2 Agent 1 has to pay c 13 for the link (3,1) 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

13. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 13 The Model of One-Way Flow Networks Network Formation Game (N,v,c) N={1,2,3,…,n} v ij is the profit for agent i for being connected to agent j c ij is the cost for agent i for making a link to agent j Example of a network g The payoff π 1 (g) for agent 1 in network g is π 1 (g) = v 13 + v 14 + v 15 + v 16 - c 13 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

14. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 14 The Model of One-Way Flow Networks More generally π i (g) = ∑ j Є Ni(g) v ij – ∑ j Є Di(g) c ij where N i (g) is the set of agents that i is connected to, and where D i (g) is the set of agents that i is directly connected to. 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

15. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 15 The Model of One-Way Flow Networks More generally π i (g) = ∑ j Є Ni(g) v ij – ∑ j Є Di(g) c ij where N i (g) is the set of agents that i is connected to, and where D i (g) is the set of agents that i is directly connected to. An action for agent i is any subset S of N\{i} indicating the set of agents that i connects to directly 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

16. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 16 The Model of One-Way Flow Networks More generally π i (g) = ∑ j Є Ni(g) v ij – ∑ j Є Di(g) c ij where N i (g) is the set of agents that i is connected to, and where D i (g) is the set of agents that i is directly connected to. An action for agent i is any subset S of N\{i} indicating the set of agents that i connects to directly A network g is a Nash network if each agent i is playing a best response in terms of his individual payoff π i (g) 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

17. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 17 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6● g

18. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 18 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} Here g -i denotes the network derived from g by removing all direct links of agent i 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6● g

19. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 19 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} Here g -i denotes the network derived from g by removing all direct links of agent i 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6● g 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

20. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 20 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} Here g -i denotes the network derived from g by removing all direct links of agent i 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6● g 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6● g-3g-3

21. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 21 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} A set S that maximizes the right-hand side of above expression is called a best response for agent i to the network g

22. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 22 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} A set S that maximizes the right-hand side of above expression is called a best response for agent i to the network g In a Nash network all agents are linked to their best responses

23. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 23 A Closer Look at Nash Networks A network g is a Nash network if for each agent i π i (g) ≥ π i (g -i + {(j,i): j Є S}) for all subsets S of N\{i} A set S that maximizes the right-hand side of above expression is called a best response for agent i to the network g In a Nash network all agents are linked to their best responses If c ik > ∑ j≠i v ij for all agents k≠i, then the only best response for agent i is the empty set Ф

24. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 24 Homogeneous Costs For each agent i all links are equally expensive: c ij = c i for all j

25. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 25 Homogeneous Costs For each agent i all links are equally expensive: c ij = c i for all j Observation for Homogeneous Costs If link (j,k) exists in g, then for agent i ≠ j,k, linking with k is at least as good as linking with j i●i● k●k● ●j●j g

26. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 26 Homogeneous Costs For each agent i all links are equally expensive: c ij = c i for all j Observation for Homogeneous Costs If link (j,k) exists in g, then for agent i ≠ j,k, linking with k is at least as good as linking with j i●i● k●k● ●j●j g

27. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 27 Homogeneous Costs For each agent i all links are equally expensive: c ij = c i for all j Observation for Homogeneous Costs If link (j,k) exists in g, then for agent i ≠ j,k, linking with k is at least as good as linking with j “Downstream Efficiency” i●i● k●k● ●j●j g

28. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 28 Lemma For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

29. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 29 Lemma For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

30. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 30 Lemma For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks When removing (2,1) agent 1 looses profits from agents 2, 3, 4, 5, 6 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

31. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 31 Lemma For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks When adding (4,1) agent 1 pays an additional cost of c 14 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

32. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 32 Lemma For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks When replacing (2,1) by (4,1) agent 1 looses profits from agents 2 and 3 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

33. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 33 Theorem For any network formation game (N,v,c) with homogeneous costs, a Nash network exists

34. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 34 Theorem For any network formation game (N,v,c) with homogeneous costs, a Nash network exists Proof by induction to the number of agents:

35. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 35 Theorem For any network formation game (N,v,c) with homogeneous costs, a Nash network exists Proof by induction to the number of agents: If n=1, then the trivial network is a Nash network

36. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 36 Theorem For any network formation game (N,v,c) with homogeneous costs, a Nash network exists Proof by induction to the number of agents: If n=1, then the trivial network is a Nash network Induction hypothesis: Nash networks exist for all network games with less than n agents.

37. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 37 Theorem For any network formation game (N,v,c) with homogeneous costs, a Nash network exists Proof by induction to the number of agents: If n=1, then the trivial network is a Nash network Induction hypothesis: Nash networks exist for all network games with less than n agents. Suppose that (N,v,c) is a network game with n agents for which NO Nash network exists.

38. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 38 Recall the Lemma: For any network formation game (N,v,c) with homogeneous costs and with c i ≤ ∑ j≠i v ij for all agents i, all cycle networks are Nash networks 2●2● 1●1● 3●3● ●4●4 ●5●5 6●6●

39. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 39 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij

40. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 40 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n

41. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 41 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’

42. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 42 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis)

43. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 43 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c)

44. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 44 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c) Therefore there is an agent i for whom the links in g are no best response

45. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 45 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c) Therefore there is an agent i for whom the links in g are no best response This agent i can not be agent n so w.l.o.g. this agent is agent 1

46. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 46 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c) Therefore there is an agent i for whom the links in g are no best response This agent i can not be agent n so w.l.o.g. this agent is agent 1 and he has a best response T with n Є T

47. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 47 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c) Therefore there is an agent i for whom the links in g are no best response This agent i can not be agent n so w.l.o.g. this agent is agent 1 and he has a best response T with n Є T and therefore c 1 ≤ v in

48. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 48 Proof continued: Hence there is at least one agent i with c i > ∑ j≠i v ij W.l.o.g. this agent is agent n Consider (N’,v’,c’) with N’=N\{n} and with v and c restricted to agents in N’ Let g’ be a Nash network in (N’,v’,c’) (induction hypothesis) Then by assumption g’ is no Nash network in (N,v,c) Therefore there is an agent i for whom the links in g are no best response This agent i can not be agent n W.l.o.g. this agent is agent 1 and he has a best response T with n Є T and therefore c 1 ≤ v in 1●1● ●n●n

49. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 49 Proof continued: Now recall that for any other agent i linking to agent 1 would be at least as good as linking to agent n v ij for j≠1 Define v ij * = v i1 + v in for i≠1, j=1 v 11 + v 1n for i=1 1●1● ●n●n i●i●

50. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 50 Proof continued: Now recall that for any other agent i linking to agent 1 would be at least as good as linking to agent n v ij for j≠1 Define v ij * = v i1 + v in for i≠1, j=1 v 11 + v 1n for i=1 Now π* i (g) = π i (g + (n,1)) for any network g on N’ and for any agent i in N’ 1●1● ●n●n i●i●

51. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 51 Proof continued: Now recall that for any other agent i linking to agent 1 would be at least as good as linking to agent n v ij for j≠1 Define v ij * = v i1 + v in for i≠1, j=1 v 11 + v 1n for i=1 Now π* i (g) = π i (g + (n,1)) for any network g on N’ and for any agent i in N’ The game (N’,v*,c’) has a Nash network g* 1●1● ●n●n i●i●

52. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 52 Proof continued: This network g* can not be a Nash network in (N,v,c)

53. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 53 Proof continued: This network g* can not be a Nash network in (N,v,c) Hence at least one agent is not playing a best response

54. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 54 Proof continued: This network g* can not be a Nash network in (N,v,c) Hence at least one agent is not playing a best response However, any agent improving in (N,v,c) contradicts that g* is a Nash network in (N’,v*,c’) by the way that v* and v are related to eachother

55. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 55 Proof continued: This network g* can not be a Nash network in (N,v,c) Hence at least one agent is not playing a best response However, any agent improving in (N,v,c) contradicts that g* is a Nash network in (N’,v*,c’) by the way that v* and v are related to eachother

56. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 56 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist

57. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 57 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist A heterogeneous costs structure 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε

58. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 58 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist A heterogeneous costs structure other links to agent 1 cost 1+ε 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε

59. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 59 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist A heterogeneous costs structure other links to agent 1 cost 1+ε other links to agent 2 cost 2+ε 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε

60. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 60 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist A heterogeneous costs structure other links to agent 1 cost 1+ε other links to agent 2 cost 2+ε other links to agents 3 and 4 cost 3+ε 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε

61. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 61 Example For network formation games (N,v,c) with heterogeneous costs, Nash networks do not need to exist A heterogeneous costs structure other links to agent 1 cost 1+ε other links to agent 2 cost 2+ε other links to agents 3 and 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε

62. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 62 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф.

63. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 63 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays {2}, then agent 1 plays {4}.

64. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 64 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays {2}, then agent 1 plays {4}. Then agent 2 plays {1}, because agent 3 never plays {1}.

65. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 65 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays {2}, then agent 1 plays {4}. Then agent 2 plays {1}, because agent 3 never plays {1}. Then agent 3 plays {2}.

66. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 66 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays {2}, then agent 1 plays {4}. Then agent 2 plays {1}, because agent 3 never plays {1}. Then agent 3 plays {2}. Then agent 4 should play Ф.

67. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 67 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part A) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays {2}, then agent 1 plays {4}. Then agent 2 plays {1}, because agent 3 never plays {1}. Then agent 3 plays {2}. Then agent 4 should play Ф. A contradiction

68. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 68 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф.

69. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 69 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4.

70. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 70 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4. Then agent 2 plays {1}.

71. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 71 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4. Then agent 2 plays {1}. Then agent 3 plays {2}.

72. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 72 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4. Then agent 2 plays {1}. Then agent 3 plays {2}. Then agent 1 plays {3,4}.

73. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 73 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4. Then agent 2 plays {1}. Then agent 3 plays {2}. Then agent 1 plays {3,4}. Then agent 4 should play {2}.

74. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 74 Example Explained The cost/payoff structure other links to 1 cost 1+ε other links to 2 cost 2+ε other links to 3, 4 cost 3+ε profits v ij =1 for all i and j 1●1● 2●2● 4●4●●3●3 1-ε 2-ε 3-ε The arguments (part B) In any Nash network agent 3 and agent 4 would either play {2} or Ф. If agent 4 plays Ф, then agent 1 plays S containing 4. Then agent 2 plays {1}. Then agent 3 plays {2}. Then agent 1 plays {3,4}. Then agent 4 should play {2}. Again a contradiction

75. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 75 Concluding remarks Our proof implies that for homogeneous costs Nash networks exist that contain at most one cycle and where every vertex has outdegree at most 1

76. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 76 Concluding remarks Our proof implies that for homogeneous costs Nash networks exist that contain at most one cycle and where every vertex has outdegree at most 1 Our model is based mainly on: V. Bala & S. Goyal (2000): A non-cooperative model of network formation. Econometrica 68, 1181-1229. A. Galeotti (2006): One-way flow networks: the role of heterogeneity. Economic Theory 29, 163-179.

77. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 77 Time for Questions a preprint of our paper is available upon request frank@micc.unimaas.nl

78. International Conference on Modeling, Computation and Optimization New Delhi, January 9-10, 2008 78 Thank you for your attention! a preprint of our paper is available upon request frank@micc.unimaas.nl