Modeling a case of herding behavior in a multi-player game

Abstract

The system mentioned in the title belongs to the family of the so-called massively multi-player online social games. It features a scoring system for the elements of the game that is prone to herding effects. We analyze in detail its stationary regime in the thermodynamic limit, when the number of players tends to infinity. In particular, for some classes of input sequences and selection policies, we provide necessary and sufficient conditions for the existence of a complete meanfield-like measure, showing off an interesting condensation phenomenon.

Appendix: Generators, cores and weak convergence

Here we quote the material necessary for the proof of Theorem 2.2. The results are borrowed from [3].

Theorem 6.1

If $O$ is the generator of a strongly continuous semigroup $\{T(t)\}$ on $\mathcal{L }$, then its domain $\mathcal{D }(O)$ is dense in $\mathcal{L }$ and $O$ is closed.

Definition 6.2

[3, p. 17] Let $O$ be a closed linear operator with domain $\mathcal{D }(O)$. A subspace $S$ of $\mathcal{D }(O)$ is said to be a core for $O$ if the closure of the restriction of $O$ to $S$ is equal to $O$, i.e., if $\overline{O_{|S}}=O$.

The next proposition is an important criterion to characterize a core.

Theorem 6.3

[3, p. 17] Let $O$ be the generator of a strongly continuous contraction semigroup $\{T(t)\}$ on $\mathcal{L }$. Let $\mathcal{D }_0$ and $\mathcal{D }$ be dense subspaces of $\mathcal{L }$ with $\mathcal{D }_0\subset \mathcal{D }\subset \mathcal{D }(O)$. If $T(t): D_0\rightarrow \mathcal{D }$ for all $t\ge 0$, then $\mathcal{D }$ is a core for $O$.

The proof of Theorem 2.1 relies heavily on the next general proposition.

Theorem 6.4

[3, Theorem 6.1, p. 28] In addition to $\mathcal{L }$, let $\mathcal{L }_k, k\ge 1$, be a sequence of Banach spaces, $\Pi _k : \mathcal{L }\rightarrow \mathcal{L }_k$ be a bounded linear transformation, subject to the constraint $\sup _k \Vert \Pi _k\Vert <\infty$. Let also $\{T_k(t)\}$ and $\{T(t)\}$ be strongly continuous contraction semigroups on $\mathcal{L }_k$ and $\mathcal{L }$ with generators $O_k$ and $O$. We write $f_k\rightarrow f$ to mean exactly

$$\begin{aligned} f\in \mathcal{L }, \quad f_k\in \mathcal{L }_k \quad \mathrm{{for}} \ k\ge 1, \quad \mathrm{{and}}\ \lim _{k\rightarrow \infty }\Vert f_k - \Pi _kf\Vert =0 . \end{aligned}$$

Then, if $D$ is a core for $O$, the following are equivalent:

• (a) For each $f\in \mathcal{L }$, $T_k(t)\Pi _kf\rightarrow T(t)f$ for all $t\ge 0$, uniformly on bounded intervals.

• (b) For each $f\in \mathcal{L }$, $T_k(t)\Pi _kf\rightarrow T(t)f$ for all $t\ge 0$.

• (c) For each $f\in D$, there exists $f_k\in \mathcal{D }(O_k)$ for each $k\ge 1$, such that $f_k\rightarrow f$ and $O_kf_k\rightarrow Of$.