Umberto Gostoli (2008)
A Cognitively Founded Model of the Social Emergence of Lexicon
Journal of Artificial Societies and Social Simulation
vol. 11, no. 1 2
<https://www.jasss.org/11/1/2.html>
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Received: 31May2006 Accepted: 28Sep2007 Published: 31Jan2008
(1) 
However, with the same set of words and meanings, we can build a M x W matrix of conditional probabilities P(m/w) each element of which represents the conditional probability that given a word w_{i} we can expect the meaning m_{j}. The generic element of this matrix p_{ji }is given by the formula (2):
(2) 
A speaker who uses this second conditional probability matrix, in a certain way, adopts the point of view of the hearer, asking itself what is the meaning that the hearer would expect to be associated to the word sent by the speaker. It is the socalled obverter learning introduced by Oliphant and Batali (1997).
Figure 1. A mental model 
Figure 2: A generic database  
Var. 1  Var. 2  Var. 3  Var. 4  Dep. Var. 
0  0  1  0  0 
0  0  1  1  1 
0  1  0  0  0 
...  ...  ....  ....  ... 
1  0  1  1  1 
1  1  0  0  0 
1  1  0  1  2 
....  ....  ....  ....  ... 
2  1  1  1  0 
2  0  1  0  0 
2  0  0  1  2 
Dep. Var. = W1*Var. 1 + W2*Var. 2 + W3*Var. 3 + W4*Var. 4 
where W1, W2, W3 and W4 are the weights of the independent variables.
Figure 3. The decision tree resulting from data mining analysis performed on the database shown in Figure 2 
E =  ( n _{0}/ N)log_{ r } (_{ n }_{0}/ N)  ( n _{1}/ N)log_{ r } ( n _{1}/ N)  (3) 
For example, if we look at the dependent variable column of the database of Figure 10, we see that there are 8 instances that have value 0 and 12 instances that have value 1. The entropy of this database is given by:
E(D) =  (8/20)log_{2} (8/20)  (12/20) log_{2} (12/20) 
info(D) = 1 0.971 = 0.029 
Figure 4 A database  
I1  I2  I3  V  
1  0  2  1  1 
2  0  1  0  1 
3  0  2  0  1 
4  1  2  1  0 
5  0  1  1  0 
6  1  0  1  0 
7  1  2  1  1 
8  0  0  0  1 
9  0  0  0  0 
10  1  0  1  1 
11  1  1  0  0 
12  1  1  1  0 
13  1  1  0  0 
14  1  2  0  1 
15  0  1  0  1 
16  1  1  1  0 
17  0  1  0  1 
18  0  1  0  1 
19  0  2  0  1 
20  1  0  1  1 
E[V(I1=0)] =  (2/10)log_{2} (2/10)  (8/10) log_{2} (8/10) = 0.722 E[V(I1=1)] =  (6/10)log_{2} (6/10)  (4/10) log_{2} (4/10) = 0.971 
Figure 5: Figure 5: V(I1=0) 
V(I1=0) 
1 
1 
1 
0 
1 
0 
1 
1 
1 
1 
Figure 6: Figure 6: V(I1=1) 
V(I1=1) 
0 
0 
1 
1 
0 
0 
0 
1 
0 
1 
E(I1) = (10/20) * 0.722 + (10/20) * 0.971 = 0.846 
info(I1) = 1 0.846 = 0.154 
E[V(I2=0)] =  (2/5)log_{2} (2/5)  (3/5) log_{2} (3/5) = 0.971 E[V(I2=1)] =  (5/9)log_{2} (5/9)  (4/9) log_{2} (4/9) = 0.994 E[V(I2=2)] =  (1/6)log_{2} (1/6)  (5/6) log_{2} (5/6) = 0.649

Figure 7: V(I2=0) 
V(I2=0) 
0 
1 
0 
1 
1 
Figure 8: V(I2=1) 
V(I2=1) 
1 
0 
0 
0 
0 
1 
0 
1 
1 
Figure 9: V(I2=2) 
V(I2=2) 
1 
1 
0 
1 
1 
1 
E(I2) = (5/20) * 0.971 + (9/20) * 0.994 + (6/20) * 0.649 = 0.885 
info(I2) = 1 0.885 = 0.115 
info(I3) = 1 0.912 = 0.088 
Figure 10. First node 
Figure 11: D(I1 = 0)  
I2  I3  V  
1  2  1  1 
2  1  0  1 
3  2  0  1 
4  1  1  0 
5  0  0  1 
6  0  0  0 
7  1  0  1 
8  1  0  1 
9  1  0  1 
10  2  0  1 
Figure 12: D(I1 = 1)  
I2  I3  V  
1  2  1  0 
2  0  1  0 
3  2  1  1 
4  0  1  1 
5  1  0  0 
6  1  1  0 
7  1  0  0 
8  2  0  1 
9  1  1  0 
10  0  1  1 
Figure 13: Agent Tom experiences' database  
Symbol 0  Symbol 1  Symbol 2  Number 
A  B  A  5 
B  A  A  0 
B  B  A  7 
...  ....  ....  ... 
B  A  B  2 
B  B  A  6 
A  A  A  6 
....  ....  ....  ... 
A  B  B  4 
A  B  B  3 
A  A  B  2 
Figure 14. Player S Mental Model 
Figure 15. Player R Mental Model 
Figure 16. Number of agents who coordinate successfully 
Figure 17. Number of messages chosen 
Figure 18. Average Number of Periods to reach equilibrium: the effect of the population's size 
Figure 19. Average Number of Periods to reach equilibrium: the effect of the words' number 
Figure 20. Average Number of Periods to reach equilibrium: the effect of the memory's length 
Figure 21. Average Number of Periods to reach equilibrium: the mental model updating lag 
I wish to thank my PhD supervisor Prof. Roberto Scazzieri for his constant support during all the stages of the research.
I am grateful to Prof. Nigel Gilbert who, during the threemonth research period I spent at the University of Surrey, gave me precious advice that helped me to critically reconsider my work, allowing me to amend some of its weak points.
I wish to thank Prof. Pietro Terna and his PhD students for the invaluable help they gave me regarding the technical aspects of the agentbased simulation.
import java.util.ArrayList; import java.util.Random; public class Agent { private int Address; private int Object; private int Age; private boolean Forecast; private boolean DT; private int[] Word; private int MemoryLenght; private int Period; private int[][] Sym; private int[] Meaning; private int Root; private int[] SnNode; private int[][] TrNode; private int[][][] TrNodeLeaf; private int Guess; private int numberObjects; private int wordLenght; private int numberSymbols; Random rand = new Random(); private int numAgents; private int LearningPeriod; public Agent (int i, int no, int wl, int ns, int p, int m) { rand.setSeed(Society.get_Seed()*(i+1)); Address = i; numberObjects = no; wordLenght = wl; numberSymbols = ns; numAgents = p; Period = 0; MemoryLenght = m; LearningPeriod = MemoryLenght2; Root = 1; DT = false; Sym = new int[wordLenght][MemoryLenght]; Meaning = new int[MemoryLenght]; Word = new int[wordLenght]; SnNode = new int[numberSymbols]; TrNode = new int[numberSymbols][numberSymbols]; TrNodeLeaf = new int [numberSymbols][numberSymbols][numberSymbols]; } public void build_DecisionTree() { if ( Period > LearningPeriod && rand.nextDouble() < 0.2 ) { DT = true; double[][] cs = new double[wordLenght][numberSymbols]; double[][][] csm = new double[wordLenght][numberSymbols][numberObjects]; double[] infoVal = new double[wordLenght]; int[][][][] countWord = new int[numberSymbols][numberSymbols][numberSymbols][numberObjects]; for ( int i = 0; i < numberSymbols; i++ ) { for ( int j = 0; j < numberSymbols; j++ ) { for ( int y = 0; y < numberSymbols; y++ ) { for ( int k = 0; k < numberObjects; k++ ) { countWord[i][j][y][k] = 0; }}}} for ( int j = 0; j < wordLenght; j++ ) { infoVal[j] = 0; for ( int i = 0; i < numberSymbols; i++ ) { cs[j][i] = 0; for ( int z = 0; z < numberObjects ; z++ ) { csm[j][i][z] = 0; }}} for ( int j = 0; j < wordLenght; j++ ) { for ( int y = 0; y < numberSymbols; y++ ) { for ( int i = 0; i < Period ; i++ ) { if ( Sym[j][i] == y ) cs[j][y] += 1; for ( int z = 0; z < numberObjects ; z++ ) { if ( Sym[j][i] == y && Meaning[i] == z ) csm[j][y][z] += 1; }}}} for ( int j = 0; j < wordLenght; j++ ) { for ( int i = 0; i < numberSymbols; i++ ) { for ( int z = 0; z < numberObjects ; z++ ) { if ( cs[j][i] != 0 && csm[j][i][z] != 0 ) infoVal[j] += cs[j][i]*(1*(csm[j][i][z]/cs[j][i])*Math.log(csm[j][i][z]/cs[j][i])); }}} double bestIV = infoVal[0]; for ( int j = 1; j < wordLenght; j++ ) { if ( infoVal[j] < bestIV ) { bestIV = infoVal[j]; }} int[] check2 = new int[wordLenght]; for ( int k = 0; k < wordLenght; k++ ) { check2[k] = 0; } for ( int h = 0; h < wordLenght; h++ ) { if ( infoVal[h] == bestIV ) check2[h] = 1; } int f; do { f = rand.nextInt(wordLenght); }while (check2[f] == 0); Root = f; int countSR; for ( int q = 0; q < numberSymbols; q++ ) { countSR = 0; for ( int r = 0; r < Period ; r++ ) { if ( Sym[Root][r] == q ) countSR += 1; } int[][] srSym = new int[wordLenght1][countSR]; int[] srMea = new int[countSR]; for ( int j = 0, y = 0; j < wordLenght1 && y < wordLenght; ) { int p = 0; if ( Root == y ) y += 1; for ( int i = 0; i < Period ; i++ ) { if ( Sym[Root][i] == q ) { srSym[j][p] = Sym[y][i]; srMea[p] = Meaning[i]; p += 1; } } j += 1; y += 1; } double[][] sr_cs = new double[wordLenght1][numberSymbols]; double[][][] sr_csm = new double[wordLenght1][numberSymbols][numberObjects]; double[] sr_infoVal = new double[wordLenght1]; for ( int j = 0; j < wordLenght1; j++ ) { sr_infoVal[j] = 0; for ( int i = 0; i < numberSymbols; i++ ) { sr_cs[j][i] = 0; for ( int z = 0; z < numberObjects ; z++ ) { sr_csm[j][i][z] = 0; }}} for ( int j = 0; j < wordLenght1; j++ ) { for ( int y = 0; y < numberSymbols; y++ ) { for ( int i = 0; i < countSR ; i++ ) { if ( srSym[j][i] == y ) sr_cs[j][y] += 1; for ( int z = 0; z < numberObjects ; z++ ) { if ( srSym[j][i] == y && srMea[i] == z ) sr_csm[j][y][z] += 1; }}}} for ( int j = 0; j < wordLenght1; j++ ) { for ( int i = 0; i < numberSymbols; i++ ) { for ( int z = 0; z < numberObjects ; z++ ) { if ( sr_cs[j][i] != 0 && sr_csm[j][i][z] != 0 ) sr_infoVal[j] += sr_cs[j][i]* (1*(sr_csm[j][i][z]/sr_cs[j][i])*Math.log(sr_csm[j][i][z]/sr_cs[j][i])); }}} double sr_bestIV = sr_infoVal[0]; for ( int j = 1; j < wordLenght1; j++ ) { if ( sr_infoVal[j] < sr_bestIV ) sr_bestIV = sr_infoVal[j]; } for ( int k = 0; k < wordLenght1; k++ ) { check2[k] = 0; } for ( int h = 0; h < wordLenght1; h++ ) { if ( sr_infoVal[h] == sr_bestIV ) check2[h] = 1; } do { f = rand.nextInt(wordLenght1); } while (check2[f] == 0); SnNode[q] = f; if ( Root == 0 ) { SnNode[q] += 1; } if ( Root == 1) { if ( SnNode[q] > 0 ) SnNode[q] += 1; } for ( int k = 0; k < numberSymbols; k++ ) { for ( int i = 0; i < wordLenght; i++ ) { if ( i != Root && i != SnNode[q] ) TrNode[q][k] = i; } for ( int v = 0; v < numberSymbols; v++ ) { for ( int h = 0; h < numberObjects; h++ ) { for ( int t = 0; t < Period; t++ ) { if ( Sym[Root][t] == q && Sym[SnNode[q]][t] == k && Sym[TrNode[q][k]][t] == v && Meaning[t] == h ) countWord[q][k][v][h] += 1; } int bestMeaning = 0; for ( int h = 0; h < numberObjects; h++ ) { if ( countWord[q][k][v][h] > bestMeaning ) bestMeaning = countWord[q][k][v][h]; } int[] countTNL = new int[numberObjects]; for ( int h = 0; h < numberObjects; h++ ) { countTNL[h] = 0; } for ( int h = 0; h < numberObjects; h++ ) { if ( countWord[q][k][v][h] == bestMeaning ) countTNL[h] = 1; } do { f = rand.nextInt(numberObjects); }while(countTNL[f] == 0 ); TrNodeLeaf[q][k][v] = f; }}}}} public void update_DataSet() { for ( int j = 0; j < wordLenght; j++ ) { for ( int i = Period1; i > 1 ; i ) { Sym[j][i+1] = Sym[j][i]; } } for ( int i = Period1; i > 1 ; i ) { Meaning[i+1] = Meaning[i]; } for ( int j = 0; j < wordLenght; j++ ) { Sym[j][0] = Word[j]; } Meaning[0] = Object; if ( Period < MemoryLenght1 ) Period += 1; } public void think_Word() { if ( DT == true ) { int c = 0; int[][] word = new int[numberSymbols*numberSymbols*numberSymbols][wordLenght]; for ( int i = 0; i < numberSymbols; i++ ) { for ( int j = 0; j < numberSymbols; j++ ) { for ( int k = 0; k < numberSymbols; k++ ) { if ( TrNodeLeaf[i][j][k] == Object ) { word[c][Root] = i; word[c][SnNode[i]] = j; word[c][TrNode[i][j]] = k; c += 1; }}}} if ( c != 0 ) { int f = rand.nextInt(c); for ( int j = 0; j < wordLenght; j++ ) { Word[j] = word[f][j]; } } else { for ( int j = 0; j < wordLenght; j++ ) { Word[j] = rand.nextInt(numberSymbols);; }}} else { for ( int j = 0; j < wordLenght; j++ ) { Word[j] = rand.nextInt(numberSymbols);; }}} public void draw_Object() { Object = rand.nextInt(numberObjects); } public void guess_Object() { if ( DT == true ) { int a = Word[Root]; int b = Word[SnNode[a]]; int c = Word[TrNode[a][b]]; Guess = TrNodeLeaf[a][b][c]; if ( Guess == 1 ) Guess = rand.nextInt(numberObjects); } else { Guess = rand.nextInt(numberObjects); } if ( Guess == Object ) Forecast = true; else Forecast = false; } public void get_Guess(Agent a) { Guess = a.guess(); if ( Guess == Object ) Forecast = true; else Forecast = false; } private int guess() { return Guess; } public int get_Object() { return Object; } public int[] get_Sig() { return Word; } public void get_Message(Agent a) { Object = a.get_Object(); int[] word = a.get_Sig(); for ( int j = 0; j < wordLenght; j++ ) { Word[j] = word[j]; } } public boolean get_Forecast() { return Forecast; } public int get_Word0() { return Word[0]; } public int get_Word1() { return Word[1]; } public int get_Word2() { return Word[2]; } }
import java.util.LinkedList; import java.util.Random; public class Society { private static final long Seed = 918273; private static int numAgents = 200; private static int Memory = 60; private static int numberObjects = 8; private static int wordLenght = 3; private static int numberSymbols = 2; private static int TotalPayOff = 0; private static int Time = 1; static java.util.List<number> urn; static Random rand = new Random(); public static void main(String[] args) { rand.setSeed(Seed); urn = new LinkedList<number>(); Agent[] agent = new Agent[numAgents]; for ( int i = 0; i < numAgents; i++ ) { agent[i] = new Agent(i, numberObjects, wordLenght, numberSymbols, numAgents, Memory); } for (int t = 0; t < 2000; t++) { for (int i = 0; i < numAgents; i++ ) { urn.add(new Number(i)); } for (int i = 0; i < numAgents/2; i++ ) { int f; int r; Number n; f = rand.nextInt(urn.size()); n = urn.get(f); r = n.draw_Number(); urn.remove(f); Agent a = agent[r]; f = rand.nextInt(urn.size()); n = urn.get(f); r = n.draw_Number(); urn.remove(f); Agent b = agent[r]; a.build_DecisionTree(); b.build_DecisionTree(); a.draw_Object(); a.think_Word(); b.get_Message(a); b.guess_Object(); a.get_Guess(b); a.update_DataSet(); b.update_DataSet(); } int tpo = 0; for (int i = 0; i < numAgents; i++ ) { Agent a = agent[i]; if ( a.get_Forecast() == true ) tpo += 1; } } } public static long get_Seed() { return Seed; } public static int get_Time() { return Time; } }
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