# American Institute of Mathematical Sciences

October  2017, 4(4): 285-318. doi: 10.3934/jdg.2017016

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

 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

Received  December 2016 Revised  May 2017 Published  September 2017

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.

Citation: Mark Broom, Chris Cannings. Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences. Journal of Dynamics & Games, 2017, 4 (4) : 285-318. doi: 10.3934/jdg.2017016
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##### References:
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
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.
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
 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
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
 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
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 $c_4$ $c_3$ $c_1$ $c_0$ $5$ $.3$ $.5$ $0$ $1.2$ $4^L$ $1.2$ $0$ $\underline{.3}$ $.5$ $4^R$ $.3$ $.5$ $\underline{0}$ $1.2$ $7$ $.3$ $.5$ $\underline{0}$ $1.2$ $1$ $\underline{.3}$ $.5$ $0$ $1.2$ $13$ $.5$ $.3$ $0$ $\underline{1.2}$ $21$ $.75$ *$\underline{.3125}$* $.28125$ $.65625$ $37$ $.7$ *$\underline{.3}$* $.2$ $.8$ $6$ $.3$ $.5$ $\underline{0}$ $1.2$ $53$ $.929$ *$\underline{.214}$* $.357$ $.5$
 index $c_4$ $c_3$ $c_1$ $c_0$ $5$ $.3$ $.5$ $0$ $1.2$ $4^L$ $1.2$ $0$ $\underline{.3}$ $.5$ $4^R$ $.3$ $.5$ $\underline{0}$ $1.2$ $7$ $.3$ $.5$ $\underline{0}$ $1.2$ $1$ $\underline{.3}$ $.5$ $0$ $1.2$ $13$ $.5$ $.3$ $0$ $\underline{1.2}$ $21$ $.75$ *$\underline{.3125}$* $.28125$ $.65625$ $37$ $.7$ *$\underline{.3}$* $.2$ $.8$ $6$ $.3$ $.5$ $\underline{0}$ $1.2$ $53$ $.929$ *$\underline{.214}$* $.357$ $.5$
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 0L 1.200000 0.000000 0.500000 0.300000 0R 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 4L 1.200000 0.000000 0.300000 0.500000 4R 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 8L 1.200000 0.000000 0.500000 0.300000 8R 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 12L 1.200000 0.000000 0.300000 0.500000 12R 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
 code c4 c3 c1 c0 0L 1.200000 0.000000 0.500000 0.300000 0R 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 4L 1.200000 0.000000 0.300000 0.500000 4R 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 8L 1.200000 0.000000 0.500000 0.300000 8R 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 12L 1.200000 0.000000 0.300000 0.500000 12R 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
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
 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
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
 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
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 *
 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 *
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) * * * * * * * * * * $[.5, .5]$
 $t_{04} \backslash t_{13}$ $.2$ $.214$ $.281$ $.285$ $.300$ $.3125$ $.318$ $.375$ $.388$ $.5$ $1.2$ * * * * $S_1$ * * * * $S_2$ .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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