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The Prisoner's Dilemma is used as a model in processes involving reciprocity; however, its classical setup can be insufficient in settings where the symmetry of the simultaneous decision making is broken -- for example, in donor and recipient processes. In the alternating Prisoner's Dilemma model the two players take turns choosing their strategy. Assuming a finite memory setup, we establish the mathematical aspects of the adaptive dynamics of the alternating Prisoner's Dilemma, paying particular attention to the case of memory 1.

We improve the solution of the classical prisoners and drawers riddle, where all prisoners can find their number using the pointer-following strategy, provided that the prisoners can send a spy to inspect all drawers and swap one pair of numbers. In the traditional approach, each prisoner may need to open up to half of the drawers. We show that this strategy is sub-optimal. Remarkably, a single swap allows all $n$ prisoners to find their number by opening only $\frac{n \ln \ln n}{\ln n} (1 + o(1))$ drawers in the worst case. We show that no strategy can do better than that by a factor larger than two. Efficiently constructing such a strategy is harder, but we provide an explicit efficient strategy that requires opening only $O(\frac{n \log \log n}{\log n})$ drawers by each prisoner in the worst case.

Recent studies in the spatial prisoner's dilemma games with reinforcement learning have shown that static agents can learn to cooperate through a diverse sort of mechanisms, including noise injection, different types of learning algorithms and neighbours' payoff knowledge. In this work, using an independent multi-agent Q-learning algorithm, we study the effects of dilution and mobility in the spatial version of the prisoner's dilemma. Within this setting, different possible actions for the algorithm are defined, connecting with previous results on the classical, non-reinforcement learning spatial prisoner's dilemma, showcasing the versatility of the algorithm in modeling different game-theoretical scenarios and the benchmarking potential of this approach. As a result, a range of effects is observed, including evidence that games with fixed update rules can be qualitatively equivalent to those with learned ones, as well as the emergence of a symbiotic mutualistic effect between populations that forms when multiple actions are defined.

Prisoners of Nazi concentration camps created paintings as a means to express their daily life experiences and feelings. Several thousand such paintings exist, but a quantitative analysis of them has not been carried out. We created an extensive dataset of 1,939 Holocaust prisoner artworks, and we employed an object detection framework that found 19,377 objects within these artworks. To support the quantitative and qualitative analysis of the art collection and its objects, we have developed an intuitive and interactive dashboard to promote a deeper engagement with these visual testimonies. The dashboard features various visual interfaces, e.g., a word cloud showing the detected objects and a map of artwork origins, and options for filtering. We presented the interface to domain experts, whose feedback highlights the dashboard's intuitiveness and potential for both quantitative and qualitative analysis while also providing relevant suggestions for improvement. Our project demonstrates the benefit of digital methods such as machine learning and visual analytics for Holocaust remembrance and educational purposes.

We investigate some versions of the famous 100 prisoner problem for the infinite case, where there are infinitely many prisoners and infinitely many boxes with labels. In this case, many questions can be asked about the admissible steps of the prisoners, the constraints they have to follow and also about the releasing conditions. We will present and analyze many versions and cases. In the infinite case, the solutions and methods require mainly analysis rather than combinatorics.

We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large number of times. The rooms are indistinguishable, except that each one has $s$ lightswitches; the prisoners win their freedom if at some point a prisoner can correctly declare that each prisoner has been in every room at least once. What is the minimum number of switches per room, $s$, such that the prisoners can manage this? We show that if the prisoners do not know the switches' starting configuration, then they have no chance of escape -- but if the prisoners do know the starting configuration, then the minimum sufficient $s$ is surprisingly small. The analysis gives rise to a number of puzzling open questions, as well.

In this paper I present a mathematically novel approach to the Prisoner's Dilemma. I do so by first defining recursively a distinct action type, what I call 'universalizing', that I add to the original prisoner's dilemma. Such a modified version of the Prisoner's Dilemma provides a very food productive model of the choices that would be made in a prisoner's dilemma by agents who trust each other. As I show, players playing a universalized prisoner's dilemma get as far out of the dilemma as is mathematically possible. I then add the concept of risk to the universalized version of prisoner's dilemma. Doing so provide a model that is sensitive to the trustworthiness of the agents in any prisoner's dilemma. As I show, with no risk, agents get out of the prisoners dilemma; and with maximal risk, the succumb to it. succumb to it.

This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.

The choice of a unique Nash equilibrium (NE) is crucial in theoretical classical and quantum games. The Eiswer-Wilkens-Lewenstein quantization scheme solves the prisoner's dilemma only for high entanglement. At medium entanglement, there are multiple NEs. We investigate the selection of a unique NE in the quantum prisoner's dilemma with variable dilemma strength parameters. The risk-dominance criterion is used. The influence of the dilemma strength parameters and entanglement is emphasized. We found that entanglement completely controls the risk-dominant equilibrium. Entanglement promotes quantum-cooperation in the risk-dominant equilibrium and thus improves its outcome.

We introduce a two-player game in which one and his/her opponent attempt to pack as many ``prisoners'' as possible on the squares of an n-by-n checkerboard; each prisoner has to be ``protected'' by at least as many guards as the number of the other prisoners adjacent. Initially, the board is covered entirely with guards. The players take turns adjusting the board configuration using one of the following rules in each turn: I. Replace one guard with a prisoner of the player's color. II. Replace one prisoner of either color with a guard and replace two other guards with prisoners of the player's color. We analyze winning strategies for small n (n<5) and the maximum number of prisoners in general. We show that this maximum is less than (7n^2+4n)/11 and conjecture it is more likely 3n^2/5+O(n).

This paper addresses a mathematically tractable model of the Prisoner's Dilemma using the framework of active inference. In this work, we design pairs of Bayesian agents that are tracking the joint game state of their and their opponent's choices in an Iterated Prisoner's Dilemma game. The specification of the agents' belief architecture in the form of a partially-observed Markov decision process allows careful and rigourous investigation into the dynamics of two-player gameplay, including the derivation of optimal conditions for phase transitions that are required to achieve certain game-theoretic steady states. We show that the critical time points governing the phase transition are linearly related to each other as a function of learning rate and the reward function. We then investigate the patterns that emerge when varying the agents' learning rates, as well as the relationship between the stochastic and deterministic solutions to the two-agent system.

Game theory is fundamental to understanding cooperation between agents. Mainly, the Prisoner's Dilemma is a well-known model that has been extensively studied in complex networks. However, although the emergence of cooperation has been investigated before, the influence of memory in its evolution is not well understood. This paper presents a detailed study of cooperation dynamics in which agents have memory. We simulate the evolutionary Prisoner's dilemma game on random, scale-free and networks presenting degree-degree correlation. Through extensive simulations, we show that assortativity can improve cooperation when the temptation to defect increases. Moreover, we show that the inclusion of memory decreases the network structure influence. Our results contribute to understanding the role of the network structure and the player's memory of cooperation.

The Prisoner's Dilemma game has a long history stretching across the social, biological, and physical sciences. In 2012, Press and Dyson developed a method for analyzing the mapping of the 8-dimensional strategy profile onto the 2-dimensional payoff space in an infinitely iterated Prisoner's Dilemma game, based on Markov chain analysis and memory-one strategies. We generalize this approach and introduce the concept of strategy parameter to show that linear relations among player payoffs are a ubiquitous feature of the infinitely iterated Prisoner's Dilemma game. Our extended analysis is applied to various strategy profiles including tit-for-tat, win-stay-lose-shift, and other randomized strategy sets. Strategy profiles are identified that map onto the vertices, edges, and interior of the Prisoner's Dilemma quadrilateral in the 2-dimensional payoff (score) space. A DaMD strategy is defined based solely on "Defection after Mutual Defection" and leads to linear relations between player scores using strategy parameter analysis. The DaMD strategy is shown to result in an equal (reciprocal) or larger (extortive) score for its user compare to the other player, independent of the strategy

This manuscript explores the research topics and collaborative behaviour of authors in the field of the Prisoner's Dilemma using topic modeling and a graph theoretic analysis of the co-authorship network. The analysis identified five research topics in the Prisoner's Dilemma which have been relevant over the course of time. These are human subject research, biological studies, strategies, evolutionary dynamics on networks and modeling problems as a Prisoner's Dilemma game. Moreover, the results demonstrated the Prisoner's Dilemma is a field of continued interest, and that it is a collaborative field compared to other game theoretic fields. The co-authorship network suggests that authors are focused on their communities and that not many connections across the communities are made. The most central authors of the network are the authors connected to the main cluster. Through examining the networks of topics, it was uncovered that the main cluster is characterised by the collaboration of authors in a single topic. These findings add to the bibliometrics study in another field and present new questions and avenues of research to understand the reasons for the measured behaviours.

Changes in payoffs can transform Prisoner's Dilemma and other social dilemmas into harmonious win-win games. Using the Robinson-Goforth topology of 2x2 games, this paper analyzes how payoff swaps turn Prisoner's Dilemma into other games, compares Prisoner's Dilemmas with other families of games, traces paths that affect the difficulty of transforming Prisoner's Dilemma and other social dilemmas into win-win games, and shows how ties connect simpler and more complex games. Charts illustrate the relationships between the 144 strict ordinal 2x2 games, the 38 symmetric 2x2 ordinal games with and without ties, and the complete set of 1,413 2x2 ordinal games. Payoffs from the symmetric ordinal 2x2 games combine to form asymmetric games, generating coordinates for a simple labeling scheme to uniquely identify and locate all asymmetric ordinal 2x2 games. The expanded topology elegantly maps relationships between 2x2 games with and without ties, enables a systematic understanding of the potential for transformations in social dilemmas and other strategic interactions, offers a tool for institutional analysis and design, and locates a variety of interesting games for further research.

Recognise that people have many, possibly conflicting, aspects to their personality. We hypothesise that each separate characteristic of a personality may be treated as an independent player in a non-zero sum many player game. This idea is applied to the two person Prisoners' Dilemma as an introductory example. We assume each prisoner has a ``mercenary'' characteristic as well as an ``altruistic'' characteristic, and find that all Nash equilibria of the Prisoners' Dilemma has each prisoner in an internal conflict between their two characteristics. The hypothesis that people are composed of more than one ``player'' may explain some of the anomalies that occur in human experiments exploring game theory.

We introduce the stochastic Network-Iterated Prisoner's Dilemma (NIPD) model, a network of players playing the Prisoner's Dilemma with their neighbours, each with a memory-one strategy which they constantly and locally update to improve their success. This process is non-deterministic, and mirrors societal interactions in many relevant aspects. We use it to assess the flexibility, noise tolerance and real-world adaptability of some well-known strategies. Furthermore, in the model a new strategy naturally emerges which proves way more successful than those. We also derive some theoretical parameters that gauge the success of a strategy in this context.

Two-player games have had a long and fruitful history of applications stretching across the social, biological, and physical sciences. Most applications of two-player games assume synchronous decisions or moves even when the games are iterated. But different strategies may emerge as preferred when the decisions or moves are sequential, or the games are iterated. Zero-determinant strategies developed by Press and Dyson are a new class of strategies that have been developed for synchronous two-player games, most notably the iterated prisoner's dilemma. Here we apply the Press-Dyson analysis to sequential or asynchronous two-player games. We focus on the asynchronous prisoner's dilemma. As a first application of the Press-Dyson analysis of the asynchronous prisoner's dilemma, tit-for-tat is shown to be an efficient defense against extortionate zero-determinant strategies. Nice strategies like tit-for-tat are also shown to lead to Pareto optimal payoffs for both players in repeated prisoner's dilemma.

In the quantum version of prisoners' dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, there exist Nash equilibria that coincide with the Pareto optimal choice where both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure.