The paper considers the following model of an \(n\)-person noncooperative game \(G\) related to network formation: There is a set \(N=\{1,\ldots, n\}\) of the players. A strategy \(g_i\) of each player \(i\in N\) is to chose a subset \(S(g_i)\subset N\setminus i\) and initiate directed links from all the players from \(S\) to player \(i\).
After choosing strategies \(g=(g_1,\ldots,g_n)\) by all the players, network relations among the players are formally represented by a directed graph \((N,E(g))\) whose nodes are the players. For every multistrategy \(g\), it is assumed that player \(i\) receives information of value \(V_{ij}>0\) from player \(j\) if there is a path from \(j\) to \(i\) in \((N,E(g))\), and \(V_{ij}=0\), otherwise. Further, it is assumed that player \(i\) initiating a link to any player \(j\in S(g_i)\), incurs a cost \(c_{ij}\). Using this, for a multistrategy \(g=(g_1,\ldots,g_n)\) the payoff function of player \(i\in N\) in game \(G\) is defined as \(\pi_i(g) = \sum_{j\in N\setminus i}V_{ij} - \sum_{j\in S(g_i)}c_{ij}\).
In the paper the authors widely discuss some general assumptions on values \(V_{ij}\) and \(c_{ij}\) (called heterogeneity by values and by players) and show when they guarantee the existence of a Nash equilibrium in game \(G\).