Network games of incomplete information represent strategic network formation when agents do not know in advance their neighbors, i.e. the network structure and the value stemming from forming links with neighboring agents. In such a setting, agents have prior beliefs about the value of attaching to their neighbors; take their action based on their prior belief and update their belief based on the history of the game. While games with a fully known network structure are widely applicable, there are many applications when players act without fully knowing with whom they interact or what their neighbors’ action will be. For example, people choosing major in college can be formalized as a network game with imperfect information: they might know something about the number of people taking that major and might infer something about the job market for different majors, but they don't know with whom they will have to interact, thus they do not know the structure of the network.
Game theoretic formulation In this setting, players have private and incomplete information about the network and this private information is interpreted as player's own type (here, private knowledge of own degree). Conditional on their own degree, players form beliefs about the degrees of their neighbors. The equilibrium concept of this game is Bayesian Nash Equilibrium.The strategy of a player is a mapping from the player's degree to the player's action. Let σ ( d ) ∈ [ 0 , 1 ] {\displaystyle \textstyle \sigma _{(d)}\in [0,1]} be the probability that a player of degree d chooses action 1. For most degrees (d) the action will be either 0 or 1, but in some cases mixed strategy might occur. The degrees of i's neighbor are drawn from a degree distribution P ~ {\displaystyle \textstyle {\tilde {P}}} , where P ~ ( d ) = P ( d ) d ⟨ d ⟩ {\displaystyle \textstyle {\tilde {P}}(d)={\frac {P(d)d}{\langle d\rangle }}} approximates the distribution over a neighbors' degree from the configuration model with respect to a degree sequence represented by P. Given P ~ {\displaystyle \textstyle {\tilde {P}}} , the probability that a neighbor takes action 1 is:
p σ = ∑ d σ ( d ) P ~ ( d ) {\displaystyle \textstyle p_{\sigma }=\sum _{d}\sigma (d){\tilde {P}}(d)} . Asymptotically, the belief that exactly m out of the d neighbors of player i choose action 1 follows a binomial distribution ( d i m ) p σ m ( 1 − p σ ) ( d i − m ) {\displaystyle \textstyle \ {d_{i} \choose m}p_{\sigma }^{m}(1-p_{\sigma })^{(d_{i}-m)}} . Thus, the expected utility of player i of degree d i {\displaystyle \textstyle \ d_{i}} who takes action x i {\displaystyle \textstyle \ x_{i}} is given by:
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