Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences

Mark Broom; Chris Cannings

doi:10.3934/jdg.2017016

Article Contents

2017, Volume 4, Issue 4: 285-318. Doi: 10.3934/jdg.2017016

This issue Previous Article Next Article Stability of the replicator dynamics for games in metric spaces

Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences

Mark Broom^1, , and
Chris Cannings^2,

1.
Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK
2.
School of Mathematics and Statistics, The University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK

* Corresponding author: Mark Broom
* Corresponding author: Mark Broom

Received: December 2016

Revised: May 2017

Published: November 2017

Abstract Full Text(HTML) Figure(2) / Table(8) Related Papers Cited by

Abstract

Animal (and human) populations contain a finite number of individuals with social and geographical relationships which evolve over time, at least in part dependent upon the actions of members of the population. These actions are often not random, but chosen strategically. In this paper we introduce a game-theoretical model of a population where the individuals have an optimal level of social engagement, and form or break social relationships strategically to obtain the correct level. This builds on previous work where individuals tried to optimise their number of connections by forming or breaking random links; the difference being that here we introduce a truly game-theoretic version where they can choose which specific links to form/break. This is more realistic and makes a significant difference to the model, one consequence of which is that the analysis is much more complicated. We prove some general results and then consider a single example in depth.

Keywords:

Mathematics Subject Classification: Primary: 05C57, 91A43; Secondary: 05C81.

Citation:

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Figure 1. 1A: Possible sequences of set membership. 1B: Schematic of the transition graph for the minimal graphs for target $43210$. For each graph, the top symbol represents the vertex with target 2, which is always a neutral vertex. The vertices with targets 0, 1, 3, 4 are represented by the symbols in the bottom left, bottom right, middle right and middle left positions, respectively. Each graph contains a specific set of links between the symbols, and the corresponding breaker or joiner status is given by the appropriate symbol. Possible transitions are shown by the arrows between the graphs

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Figure 2. The transitions starting from matrices $4$ and $8$ on the $5$-cube. The indices of the vertices and of the edges have two numbers, corresponding to the matrix reached from matrix $4$ and $8$ respectively. For example matrices 22 can be reached from matrix 4 by 16 then 2 (see Table 5). From 22 one can reach 18, 23 and 54, and 22 can be reached from 6, 20 and 30. The possible transitions from 26 are 18, 23 and 58 and can be reached from 10, 24 and 30. Stable matrices are highlighted. 2A: cost=0, 2B: cost=0.1.

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Table 1. The set membership of vertices for targets with $n=3$ and $n=4$, and number of graphs in the minimal set. The omitted sequences are all duals of those included.

Target	Sets	Min. score	Number of states
2 2 2	d d d	0	1
2 2 1	b b c	1	3
2 2 0	b b c	2	4
2 1 1	d d d	0	1
2 1 0	b d c	1	2
1 1 1	a a a	1	6
3 3 3 3	d d d d	0	1
3 3 3 2	b b b c	1	4
3 3 3 1	b b b c	2	7
3 3 3 0	b b b c	3	8
3 3 2 2	d d d d	0	1
3 3 2 1	b b d c	1	3
3 3 2 0	b b d c	2	4
3 3 1 1	b b c c	2	9
3 3 1 0	b b c c	3	12
3 3 0 0	b b c c	4	16
3 2 2 2	b a a a	1	9
3 2 2 1	d d d d	0	1
3 2 2 0	b d d c	1	2
3 2 1 1	b b c c	1	5
3 2 1 0	b b c c	2	8
3 1 1 1	d d d d	0	1
2 2 2 2	d d d d	0	1
2 2 2 1	a a a a	1	13
2 2 1 1	d d d d	0	1

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Table 2. The stationary distributions over the eight graphs $G_{1}-G_{8}$ for the 64 matrices denoted by 0-63

vector		codes		of		matrices
(2, 3, 1, 4, 0, 0, 0, 0)	0	8	16	24	32	40	48	56
(0, 0, 0, 0, 4, 1, 3, 2)	0	1	2	3	4	5	6	7
(0, 0, 0, 0, 4, 3, 1, 2)	8	9	10	11	12	13	14	15
(2, 1, 3, 4, 0, 0, 0, 0)	4	12	20	28	36	44	52	60
(3, 3, 0, 3, 1, 1, 3, 2)	17
(4, 3, 2, 2, 2, 2, 3, 4)	18
(6, 3, 0, 0, 4, 4, 9, 8)	19	23
(6, 3, 3, 6, 2, 2, 6, 4)	21
(6, 3, 6, 6, 2, 2, 3, 4)	22
(1, 1, 0, 1, 1, 1, 1, 1)	25
(4, 3, 2, 2, 6, 6, 3, 6)	26
(2, 1, 0, 0, 4, 4, 3, 4)	27	31
(2, 1, 1, 2, 2, 2, 2, 2)	29
(2, 1, 2, 2, 2, 2, 1, 2)	30
(1, 2, 0, 2, 2, 0, 2, 1)	33
(2, 3, 1, 1, 3, 0, 3, 3)	34
(1, 2, 0, 0, 4, 0, 4, 3)	35	39
(1, 1, 1, 2, 2, 0, 2, 1)	37
(1, 1, 1, 1, 1, 0, 1, 1)	38
(1, 2, 0, 2, 2, 1, 1, 1)	41
(4, 6, 2, 2, 6, 3, 3, 6)	42
(1, 2, 0, 0, 4, 2, 2, 3)	43	47
(1, 1, 1, 2, 2, 1, 1, 1)	45
(2, 2, 2, 2, 2, 1, 1, 2)	46
(3, 4, 0, 4, 0, 0, 2, 1)	49	57
(8, 9, 4, 4, 0, 0, 3, 6)	50	58
(1, 1, 0, 0, 0, 0, 1, 1)	51	55	59	63
(3, 2, 2, 4, 0, 0, 2, 1)	53	61
(4, 3, 4, 4, 0, 0, 1, 2)	54	62

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Table 3. Possible moves and outcomes for matrix 5. The first column gives the possible stationary distribution switched to, the other columns the corresponding costs for each vertex. The important cost (underlined) is that to the vertex that can make the switch. A switch can occur in the three cases highlighted by ^*s.

index	$c_4$	$c_3$	$c_1$	$c_0$
$5$	$.3$	$.5$	$0$	$1.2$
$4^L$	$1.2$	$0$	$\underline{.3}$	$.5$
$4^R$	$.3$	$.5$	$\underline{0}$	$1.2$
$7$	$.3$	$.5$	$\underline{0}$	$1.2$
$1$	$\underline{.3}$	$.5$	$0$	$1.2$
$13$	$.5$	$.3$	$0$	$\underline{1.2}$
$21$	$.75$	^$\underline{.3125}$^	$.28125$	$.65625$
$37$	$.7$	^$\underline{.3}$^	$.2$	$.8$
$6$	$.3$	$.5$	$\underline{0}$	$1.2$
$53$	$.929$	^$\underline{.214}$^	$.357$	$.5$

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Table 4. The costs for each of the individuals $t_{0}-t_{4}$, where the cost for $t_{i}$ is denoted by $c_{i}$

code	c₄	c₃	c₁	c₀
0^L	1.200000	0.000000	0.500000	0.300000
0^R	0.30000	0.500000	0.000000	1.200000
1	0.300000	0.500000	0.000000	1.200000
2	0.300000	0.500000	0.000000	1.200000
3	0.300000	0.500000	0.000000	1.200000
4^L	1.200000	0.000000	0.300000	0.500000
4^R	0.300000	0.500000	0.00000	1.200000
5	0.300000	0.500000	0.000000	1.200000
6	0.300000	0.500000	0.000000	1.200000
7	0.300000	0.500000	0.000000	1.200000
8^L	1.200000	0.000000	0.500000	0.300000
8^R	0.500000	0.300000	0.000000	1.200000
9	0.500000	0.300000	0.000000	1.200000
10	0.500000	0.300000	0.000000	1.200000
11	0.500000	0.300000	0.000000	1.200000
12^L	1.200000	0.000000	0.300000	0.500000
12^R	0.500000	0.300000	0.000000	1.200000
13	0.500000	0.300000	0.000000	1.200000
14	0.500000	0.300000	0.000000	1.200000
15	0.500000	0.300000	0.000000	1.200000
16	1.200000	0.000000	0.500000	0.300000
17	0.750000	0.312500	0.375000	0.562500
18	0.681818	0.318182	0.318182	0.681818
19	0.441176	0.500000	0.264706	0.794118
20	1.200000	0.000000	0.300000	0.500000
21	0.750000	0.312500	0.281250	0.656250
22	0.843750	0.218750	0.281250	0.656250
23	0.441176	0.500000	0.264706	0.794118
24	1.200000	0.000000	0.500000	0.300000
25	0.714286	0.285714	0.285714	0.714286
26	0.656250	0.281250	0.218750	0.843750
27	0.500000	0.388889	0.166667	0.944444
28	1.200000	0.000000	0.300000	0.500000
29	0.714286	0.285714	0.214286	0.785714
30	0.785714	0.214286	0.214286	0.785714
31	0.500000	0.388889	0.166667	0.944444
32	1.200000	0.000000	0.500000	0.300000
33	0.700000	0.300000	0.300000	0.700000
34	0.562500	0.375000	0.312500	0.750000
35	0.357143	0.500000	0.214286	0.928571
36	1.200000	0.000000	0.300000	0.500000
37	0.700000	0.300000	0.200000	0.800000
38	0.714286	0.285714	0.285714	0.714286
39	0.357143	0.500000	0.214286	0.928571
40	1.200000	0.000000	0.500000	0.300000
41	0.800000	0.200000	0.300000	0.700000
42	0.656250	0.281250	0.312500	0.750000
43	0.500000	0.357143	0.214286	0.928571
44	1.200000	0.000000	0.300000	0.500000
45	0.800000	0.200000	0.200000	0.800000
46	0.785714	0.214286	0.285714	0.714286
47	0.500000	0.357143	0.214286	0.928571
48	1.200000	0.000000	0.500000	0.300000
49	0.928571	0.214286	0.500000	0.357143
50	0.794118	0.264706	0.500000	0.441176
51	0.500000	0.500000	0.500000	0.500000
52	1.200000	0.000000	0.300000	0.500000
53	0.928571	0.214286	0.357143	0.500000
54	0.944444	0.166667	0.388889	0.500000
55	0.500000	0.500000	0.500000	0.500000
56	1.200000	0.000000	0.500000	0.300000
57	0.928571	0.214286	0.500000	0.357143
58	0.794118	0.264706	0.500000	0.441176
59	0.500000	0.500000	0.500000	0.500000
60	1.200000	0.000000	0.300000	0.500000
61	0.928571	0.214286	0.357143	0.500000
62	0.944444	0.166667	0.388889	0.500000
63	0.500000	0.500000	0.500000	0.500000

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Table 5. The optimal moves from matrices 0 to 31 (first nine columns) and matrices 32-63 (final nine columns) for cases 1a and 1b (results are identical for the two cases). The first column indicates the starting matrix, the next six the moves from the six graphs where changes can be made (listed in increasing numerical order), and the last two columns possible moves for $t_{4}$ and $t_{0}$ when both changes are allowed. We note that only one change is possible at each point, and if making no change is optimal, we simply write the starting matrix index.

0	1	2	0	0	16	32	3	48	32	33	34	32	32	32	32	35	32
1	1	1	1	1	17	33	1	49	33	33	35	33	33	49	33	33	33
2	2	2	2	2	18	34	2	50	34	35	34	34	34	50	34	33	18
3	3	3	3	3	3	3	3	3	35	35	35	35	35	35	35	35	35
4	5	6	4	4	20	36	7	52	36	37	38	36	36	36	36	39	36
5	5	5	5	5	21	37	5	53	37	37	37	37	37	53	37	37	37
6	6	6	6	6	22	38	6	54	38	39	38	34	38	54	38	37	22
7	7	7	7	7	7	7	7	7	39	39	37	39	39	39	39	39	39
8	9	10	8	8	24	40	11	56	40	41	42	40	40	40	40	43	40
9	9	9	9	9	25	41	9	57	41	41	43	41	41	41	41	41	41
10	10	10	10	10	26	42	10	58	42	43	42	42	42	58	42	41	42
11	11	11	11	11	11	11	11	11	43	43	43	43	43	43	11	43	43
12	13	14	12	12	28	44	15	60	44	45	46	44	44	44	44	47	44
13	13	13	13	13	29	45	13	61	45	45	45	45	45	45	45	45	45
14	14	14	14	14	30	46	14	62	46	47	46	42	46	62	46	45	46
15	15	15	15	15	15	15	15	15	47	47	45	47	47	47	15	47	47
16	17	18	16	16	16	16	19	16	48	48	48	48	48	48	48	48	48
17	17	19	17	17	17	49	18	33	49	49	49	49	49	49	49	49	49
18	19	18	18	18	18	50	18	18	50	50	50	50	50	50	50	50	50
19	19	19	19	19	19	19	19	19	51	51	51	51	51	51	51	51	51
20	21	22	20	20	20	20	23	20	52	52	52	52	52	52	52	52	52
21	21	23	21	21	21	53	21	37	53	52	53	53	53	53	53	53	53
22	23	22	18	22	22	54	22	22	54	54	52	50	54	54	54	53	54
23	23	23	23	23	23	23	23	23	55	54	53	55	55	55	55	52	55
24	25	26	24	24	24	24	27	24	56	56	56	56	56	56	56	56	56
25	25	27	25	17	25	57	26	41	57	57	57	57	57	41	57	57	57
26	27	26	26	18	26	58	26	26	58	58	58	58	58	58	58	58	58
27	27	27	27	19	11	27	27	43	59	59	59	59	59	43	27	59	11
28	29	30	28	28	28	28	31	28	60	60	60	60	60	60	60	60	60
29	29	31	29	21	29	61	29	45	61	60	61	61	61	45	61	61	61
30	31	30	26	22	30	62	30	30	62	62	60	58	62	62	62	61	62
31	31	31	31	23	15	31	31	47	63	62	61	63	63	47	31	60	15

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Table 6. The optimal moves from matrices 0 to 31 (first nine columns) and matrices 32-63 (final nine columns) for cases 2a and 2b (results are identical for the two cases). The first column indicates the starting matrix, the next six the moves from the six graphs where changes can be made (listed in increasing numerical order), and the last two columns possible moves for $t_{4}$ and $t_{0}$ when two changes are allowed

0	1	2	4	8	16	32	3	48	32	33	34	36	40	48	0	35	16
1	0	3	5	9	17	33	2	49	33	33	35	37	41	49	33	33	33
2	3	0	6	10	18	34	1	50	34	35	34	34	42	50	34	33	18
3	2	1	7	11	19	35	0	51	35	35	35	39	43	51	3	35	19
4	5	6	0	12	20	36	7	52	36	37	38	32	44	52	4	39	20
5	4	7	1	13	21	37	6	53	37	37	37	33	45	53	37	37	37
6	7	4	2	14	22	38	5	54	38	39	38	34	46	54	38	37	22
7	6	5	3	15	23	39	4	55	39	39	37	35	47	55	7	39	23
8	9	10	12	0	24	40	11	56	40	41	42	44	32	56	8	43	24
9	8	11	13	1	25	41	10	57	41	41	43	45	33	41	41	41	41
10	11	8	14	2	26	42	9	58	42	43	42	42	34	58	42	41	26
11	10	9	15	3	11	11	8	11	43	43	43	47	35	43	11	43	43
12	13	14	8	4	28	44	15	60	44	45	46	40	36	60	12	47	28
13	12	15	9	5	29	45	14	61	45	45	45	41	37	45	45	45	45
14	15	12	10	6	30	46	13	62	46	47	46	42	38	62	46	45	30
15	14	13	11	7	15	15	12	15	47	47	45	43	39	47	15	47	47
16	17	18	20	24	0	48	19	32	48	49	50	52	56	32	16	51	0
17	17	19	21	17	17	49	18	33	49	48	51	53	57	49	49	50	49
18	19	18	18	18	18	50	18	18	50	51	48	50	58	50	50	49	50
19	19	19	23	19	3	51	19	35	51	50	49	55	59	35	19	48	3
20	21	22	16	28	4	52	23	36	52	52	52	48	60	36	20	52	4
21	21	23	17	21	21	53	22	37	53	52	53	49	61	53	53	53	53
22	23	22	18	22	22	54	21	22	54	54	52	50	62	54	54	53	54
23	23	23	19	23	7	55	23	39	55	54	53	51	63	39	23	52	7
24	25	26	28	16	8	56	27	40	56	57	58	60	48	40	24	59	8
25	25	27	29	17	25	57	26	41	57	56	59	61	49	41	57	58	57
26	27	26	26	18	26	58	26	42	58	59	56	58	50	58	58	57	58
27	27	27	31	19	11	27	27	43	59	58	57	63	51	43	27	56	11
28	29	30	24	20	12	60	31	44	60	60	60	56	52	44	28	60	12
29	29	31	25	21	29	61	30	45	61	60	61	57	53	45	61	61	61
30	31	30	26	22	30	62	29	46	62	62	60	58	54	62	62	61	62
31	31	31	27	23	15	31	31	47	63	62	61	59	55	47	31	60	15

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Table 7. The possible PNEs for model $1a$ for various costs of changing. For zero cost the PNEs are those in the bottom row. As the cost increases there are critical points when additional matrices become PNEs, until at the highest threshold of 0.5, all 63 matrices are PNEs.

Fee		New	PNEs
.5	0	4	8	*	*
.3	12	*	*	*	*
.28125	6	24	*	*	*
.2	1	5	32	40	*
.181818	2	16	*	*	*
.161932	22	26	*	*	*
.151786	25	38	*	*	*
.150326	27	31	54	62	*
.142857	55	59	63	*	*
.129464	29	30	46	*	*
.110294	17	34	*	*	*
.1	9	13	36	44	*
.098214	21	42	*	*	*
.085714	14	28	33	37	41
.057143	43	47	53	61	*
.053467	18	*	*	*	*
.01875	10	20	*	*	*
.014286	39	57	*	*	*
0	3	7	11	15	*
	19	23	35	45	*
	48	49	50	51	*
	52	56	58	60	*

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Table 8. The space of PNEs. For any point the PNEs are all those included below and to the left of that point. Set $S_1=\{8, 9, 10, 11, 12, 13, 14, 15, 20, 32, 36, 44, 52, 60\}$ and $S_2=\{0, 1, 2, 3, 4, 5, 6, 7, 16, 24, 28, 40, 48, 56\}$.

$t_{04} \backslash t_{13}$	$.2$	$.214$	$.281$	$.285$	$.300$	$.3125$	$.318$	$.375$	$.388$	$.5$
$1.2$	*	*	*	*	$S_1$	*	*	*	*	$S_2$
.9414	*	*	*	*	*	*	*	*	(27, 31, 54, 62)	*
*	*	*	*	*	*	*	*	*	$[.7\dot{2}, .3\dot{7}]]$	*
.9285	*	*	*	*	*	*	*	*	*	(35, 39, 49, 57)
*	*	*	*	*	*	*	*	*	*	$[.357, .643]$
.8437	*	*	(22, 26)	*	*	*	*	*	*	*
*	*	*	$[.75, .25]$	*	*	*	*	*	*	*
.8	(45)	*	*	*	(37, 41)	*	*	*	*	*
*	$[.8, .2]$	*	*	*	$[.75, .25]$	*	*	*	*	*
.794	*	*	*	*	*	*	*	*	*	(19, 23, 50, 58)
.*	*	*	*	*	*	*	*	*	*	$[.618, .382]$
.785	*	(30)	*	(29, 46)	*	*	*	*	*	*
*	*	$[.786.214]$	*	$[.75, .25]$	*	*	*	*	*	*
.75	*	*	*	*	*	(21, 42)	*	(17, 34)	*	*
*	*	*	*	*	*	$[.703, .297]$	*	$[.65\dot{6}, .34\dot{3}]$	*	*
.714	*	*	*	(25, 38)	*	*	*	*	*	*
*	*	*	*	$[.714, .286]$	*	*	*	*	*	*
.7	*	*	*	*	(33)	*	*	*	*	*
*	*	*	*	*	$[.7, .3]$	*	*	*	*	*
.68	*	*	*	*	*	*	(18)	*	*	*
*	*	*	*	*	*	*	$[.682, .318]$	*	*	*
.5	*	*	*	*	*	*	*	*	*	(51, 55, 59, 63)
*	*	*	*	*	*	*	*	*	*	$[.5, .5]$

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Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences

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Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences

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24	25	26	24	24	24	24	27	24	56	56	56	56	56	56	56	56	56
25	25	27	25	17	25	57	26	41	57	57	57	57	57	41	57	57	57
26	27	26	26	18	26	58	26	26	58	58	58	58	58	58	58	58	58
27	27	27	27	19	11	27	27	43	59	59	59	59	59	43	27	59	11
28	29	30	28	28	28	28	31	28	60	60	60	60	60	60	60	60	60
29	29	31	29	21	29	61	29	45	61	60	61	61	61	45	61	61	61
30	31	30	26	22	30	62	30	30	62	62	60	58	62	62	62	61	62
31	31	31	31	23	15	31	31	47	63	62	61	63	63	47	31	60	15

0	1	2	4	8	16	32	3	48	32	33	34	36	40	48	0	35	16
1	0	3	5	9	17	33	2	49	33	33	35	37	41	49	33	33	33
2	3	0	6	10	18	34	1	50	34	35	34	34	42	50	34	33	18
3	2	1	7	11	19	35	0	51	35	35	35	39	43	51	3	35	19
4	5	6	0	12	20	36	7	52	36	37	38	32	44	52	4	39	20
5	4	7	1	13	21	37	6	53	37	37	37	33	45	53	37	37	37
6	7	4	2	14	22	38	5	54	38	39	38	34	46	54	38	37	22
7	6	5	3	15	23	39	4	55	39	39	37	35	47	55	7	39	23
8	9	10	12	0	24	40	11	56	40	41	42	44	32	56	8	43	24
9	8	11	13	1	25	41	10	57	41	41	43	45	33	41	41	41	41
10	11	8	14	2	26	42	9	58	42	43	42	42	34	58	42	41	26
11	10	9	15	3	11	11	8	11	43	43	43	47	35	43	11	43	43
12	13	14	8	4	28	44	15	60	44	45	46	40	36	60	12	47	28
13	12	15	9	5	29	45	14	61	45	45	45	41	37	45	45	45	45
14	15	12	10	6	30	46	13	62	46	47	46	42	38	62	46	45	30
15	14	13	11	7	15	15	12	15	47	47	45	43	39	47	15	47	47
16	17	18	20	24	0	48	19	32	48	49	50	52	56	32	16	51	0
17	17	19	21	17	17	49	18	33	49	48	51	53	57	49	49	50	49
18	19	18	18	18	18	50	18	18	50	51	48	50	58	50	50	49	50
19	19	19	23	19	3	51	19	35	51	50	49	55	59	35	19	48	3
20	21	22	16	28	4	52	23	36	52	52	52	48	60	36	20	52	4
21	21	23	17	21	21	53	22	37	53	52	53	49	61	53	53	53	53
22	23	22	18	22	22	54	21	22	54	54	52	50	62	54	54	53	54
23	23	23	19	23	7	55	23	39	55	54	53	51	63	39	23	52	7
24	25	26	28	16	8	56	27	40	56	57	58	60	48	40	24	59	8
25	25	27	29	17	25	57	26	41	57	56	59	61	49	41	57	58	57
26	27	26	26	18	26	58	26	42	58	59	56	58	50	58	58	57	58
27	27	27	31	19	11	27	27	43	59	58	57	63	51	43	27	56	11
28	29	30	24	20	12	60	31	44	60	60	60	56	52	44	28	60	12
29	29	31	25	21	29	61	30	45	61	60	61	57	53	45	61	61	61
30	31	30	26	22	30	62	29	46	62	62	60	58	54	62	62	61	62
31	31	31	27	23	15	31	31	47	63	62	61	59	55	47	31	60	15