Download presentation

Presentation is loading. Please wait.

Published byCarli Huck Modified over 2 years ago

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

Similar presentations

Presentation is loading. Please wait....

OK

(CSC 102) Discrete Structures Lecture 14.

(CSC 102) Discrete Structures Lecture 14.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on personal health record in cloud computing Design ppt online Ppt on endangered species of animals Convert pptx to ppt online free Ppt on articles of association for social clubs Ppt on water bodies on earth Ppt on pythagoras theorem for class 10 Ppt on windows vista operating system Ppt on waxes by andrea Ppt on duty roster