Hard and Soft Preparation Sets in Boolean Games

Abstract

Abstract.A fundamental problem in game theory is the possibility of reaching equilibrium outcomes with undesirable properties, e.g., inefficiency. The economics literature abounds with models that attempt to modify games in order to avoid such undesirable properties, for example through the use of subsidies and taxation, or by allowing players to undergo a bargaining phase before their decision. In this paper, we consider the effect of such transformations in Boolean games with costs, where players control propositional variables that they can set to true or false, and are primarily motivated to seek the satisfaction of some goal formula, while secondarily motivated to minimise the costs of their actions. We adopt (pure) preparation sets (prep sets) as our basic solution concept. A preparation set is a set of outcomes that contains for every player at least one best response to every outcome in the set. Prep sets are well-suited to the analysis of Boolean games, because we can naturally represent prep sets as propositional formulas, which in turn allows us to refer to prep formulas. The preference structure of Boolean games with costs makes it possible to distinguish between hard and soft prep sets. The hard prep sets of a game are sets of valuations that would be prep sets in that game no matter what the cost function of the game was. The properties defined by hard prep sets typically relate to goal-seeking behaviour, and as such these properties cannot be eliminated from games by, for example, taxation or subsidies. In contrast, soft prep sets can be eliminated by an appropriate system of incentives. Besides considering what can happen in a game by unrestricted manipulation of players' cost function, we also investigate several mechanisms that allow groups of players to form coalitions and eliminate undesirable outcomes from the game, even when taxes or subsidies are not a possibility.

Negotiable Votes

“…where the payoff is not necessarily constant. These boolean games have been introduced to account for efforts (or costs) in performing actions (Grant et al, 2011;Wooldridge et al, 2013;Turrini, 2016;Harrenstein et al, 2014). When comparing two outcomes, a player will prefer the ones satisfying the goal, but will otherwise look at minimising the effort.…”

“…Boolean games with costs have been looked at from the point of view of incentive engineering, allowing payoffs to be manipulable, either by exogenous taxation mechanisms as in the work of (Wooldridge et al, 2013) and (Harrenstein et al, 2014), or by endogenous negotiation as in the work of (Turrini, 2016). In the exogenous setting, an external system designer can impose taxes on players' actions, by effectively influencing their decision-making towards the realisation of his or her own goal formula.…”

“…This has lead to the discovery of the existence of hard equilibria (Harrenstein et al, 2014), i.e., pure Nash-equilibria that cannot be removed by external incentives. Notice that their presence is even more frequent in endogenous settings, due to the fact that side-payments are a weaker form of manipulation than external intervention, as observed by (Turrini, 2016).…”

Simulating cardinal preferences in Boolean games: A proof technique

“…Boolean games proved to be a popular framework for modelling aspects of multiagent systems such as incentive design (Wooldridge, 2012;Endriss et al, 2011;Levit et al, 2013;Turrini, 2013;Galafassi and Bazzan, 2013;Harrenstein et al, 2014), coalition formation (Dunne et al, 2008;Bonzon et al, 2012;Popovici and Dobre, 2012), delegation (Kraus and Wooldridge, 2012) and communication (Grant et al, 2011). However, the inability of agents in Boolean games to express preferences other than { 0, 1 } was a hindrance in this setting, and a number of frameworks sought to enrich the players' preferences by description logics (Lukasiewicz and Ragone, 2009), CP-nets, (Bonzon et al, 2009c,a), modal logics (Gutierrez et al, 2013;Ågotnes et al, 2013;Harrenstein et al, 2015), fuzzy logics (Marchioni and Wooldridge, 2014), and formula weights (Bilò, 2007;Mavronicolas et al, 2007).…”

Endogenous games with goals: side-payments among goal-directed agents

“…In the terminology of[23] such outcomes would be called soft equilibria, meaning that they can be rationally removed by the appropriate use of allowed game transformations.…”