Qualitative Inquiry, Activism, the Academy, and the Infinite Game: An Introduction to the Special Issue

This is the first of two special issues on qualitative inquiry as activism. This first issue focuses upon activism and/in the academy (academic work, academic cultures, academic practices, etc.), the second on activism in the processes of research itself and activism beyond the academy, in the world. Two issues with different themes, but the overlaps and conversations between them are both obvious and significant: inquiry is part of, rooted in, the academy; inquiry and the academy are both of, and in, the world. Drawing upon the concept of the “infinite game” where, rather than being driven by the need to win and compete (the “finite game”), we argue for the collective, collaborative work of giving close, deep attention to the human, the nonhuman, and the more-than-human in order to “create and recreate our institutions,” with activism key to this work.

Example of a Finite Game with No Berge Equilibria at All

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal-form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure, nor in mixed strategies is given.

PERIODIC STRATEGIES: A NEW SOLUTION CONCEPT AND AN ALGORITHM FOR NONTRIVIAL STRATEGIC FORM GAMES

We introduce a new solution concept, called periodicity, for selecting optimal strategies in strategic form games. This periodicity solution concept yields new insight into nontrivial games. In mixed strategy strategic form games, periodic solutions yield values for the utility function of each player that are equal to the Nash equilibrium ones. In contrast to the Nash strategies, here the payoffs of each player are robust against what the opponent plays. Sometimes, periodicity strategies yield higher utilities, and sometimes the Nash strategies do, but often the utilities of these two strategies coincide. We formally define and study periodic strategies in two player perfect information strategic form games with pure strategies and we prove that every nontrivial finite game has at least one periodic strategy, with nontrivial meaning nondegenerate payoffs. In some classes of games where mixed strategies are used, we identify quantitative features. Particularly interesting are the implications for collective action games, since there the collective action strategy can be incorporated in a purely noncooperative context. Moreover, we address the periodicity issue when the players have a continuum set of strategies available.

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

Optimal Penalties for Repeat Offenders – The Role of Offence History

Within an infinite and a corresponding finite game framework we analyse intertemporal punishment for repeat offenders. The legal authority is assumed to maximize social welfare by minimizing the sum of harm from crimes and cost of punishment. We show that the time horizon considerably affects the structure of the optimal penalty scheme. In the finite game framework decreasing as well as escalating penalty schemes may be optimal. For the more appropriate infinite game framework we show three main results: First, any penalty scheme can be replaced by a (weakly) escalating penalty scheme that leads to the same criminal activity and the same social penalization cost. Second, the optimal penalty scheme is of the escalating type. Third, the socially optimal level of crime under escalating penalties may be higher than the level which would be optimal under uniform penalties.