Multiple drawing multi-colour urns by stochastic approximation

A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc, j balls of colour j (for all 1 ≤ j ≤ d). We study multiple drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.</p

Multiple drawing multi-colour urns by stochastic approximation

Abstract A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc, j balls of colour j (for all 1 ≤ j ≤ d). We study multiple drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples. </jats:p

Limit Theorems for Stochastic Approximations Algorithms With Application to General Urn Models

In the present paper we study the multidimensional stochastic approximation algorithms where the drift function h is a smooth function and where jacobian matrix is diagonalizable over C but assuming that all the eigenvalues of this matrix are in the the region Repzq ą 0. We give results on the fluctuation of the process around the stable equilibrium point of h. We extend the limit theorem of the one dimensional Robin's Monroe algorithm [MR73]. We give also application of these limit theorem for some class of urn models proving the efficiency of this method

Preferential attachment graphs with co-existing types of different fitnesses

We extend the work of Antunović et al. (2016) on competing types in preferential attachment models to include cases where the types have different fitnesses, which may be either multiplicative or additive. We show that, depending on the values of the parameters of the models, there are different possible limiting behaviours depending on the zeros of a certain function. In particular, we show the existence of choices of the parameters where one type is favoured both by having higher fitness and by the type of attachment mechanism, but the other type has a positive probability of dominating the network in the limit

Statistical test for an urn model with random multidrawing and random addition

We complete the study of the model introduced in [11]. It is a two-color urn model with multiple drawing and random (non-balanced) time-dependent reinforcement matrix. The number of sampled balls at each time-step is random. We identify the exact rates at which the number of balls of each color grows to infinity and define two strongly consistent estimators for the limiting reinforcement averages. Then we prove a Central Limit Theorem, which allows to design a statistical test for such averages.Comment: arXiv admin note: text overlap with arXiv:2102.0628

Statistical modelling of games

This thesis mainly focuses on the statistical modelling of a selection of games, namely, the minority game, the urn model and the Hawk-Dove game. Chapters 1 and 2 give a brief introduction and survey of the field. In Chapter 3, the key characteristics of the minority game are reproduced. In addition, the minority game is extended to include wealth distribution and leverage effect. By assuming that each player has initial wealth which rises and falls according to profit and loss, with the potential of borrowing and bankruptcy, we find that modelled wealth distribution may be power law distributed and leverage increases the instability of the system. In Chapter 4, to explore the effects of memory, we construct a model where agents with memories of different lengths compete for finite resources. Using analytical and numerical approaches, our research demonstrates that an instability exists at a critical memory length; and players with different memory lengths are able to compete with each other and achieve a state of co-existence. The analytical solution is found to be connected to the well-known urn model. Additionally, our findings reveal that the temperature is related to the agent's memory. Due to its general nature, this memory model could potentially be relevant for a variety of other game models. In Chapter 5, our main finding is extended to the Hawk-Dove game, by introducing the memory parameter to each agent playing the game. An assumption is made that agents try to maximise their profits by learning from past experiences, stored in their finite memories. We show that the analytical results obtained from these two games are in agreement with the results from our simulations. It is concluded that the instability occurs when agents' memory lengths reach the critical value. Finally, Chapter 6 provides some concluding remarks and outlines some potential future work

Cooperative Models of Stochastic Growth - On a class of reinforced processes with graph-based interactions

Consider a sequence of positive integer-valued random vectors denoted by ${\bf x}_n = (x_1(n),\ldots,x_N(n))$ for $n= 0,1,2,\ldots \>$. Fix ${\bf x}_0$, and given ${\bf x}_n$, choose a \emph{random} coordinate $i_{n+1} \in \{1,\ldots,N\}$. The probability that $\{i_{n+1} = i\}$ for a particular coordinate $i$ is proportional to a non-decreasing function $f_i$ of $\sum_{j =1}^N a_{ij}x_j(n)$, where $a_{ij} \geq 0$ measures how strongly $j$ cooperates with $i$. Now, on the event that $\{i_{n+1}=i\}$, update the sequence in such a way that ${\bf x}_{n+1}={\bf x}_n + {\bf e}_i$, where ${\bf e}_i$ is the vector whose $i$-th coordinate is 1 and whose other coordinates are 0. Finally, given $A=(a_{ij})_{i,j=1}^N$ and $f_i, \> i =1,\ldots,N,$ what can one say about $\lim_{n \to \infty} n^{-1} {\bf x}_n$