With a little help from your friends: semi-cooperative games via Joker moves

TL;DR

This work introduces Joker games, a variant of two-player concurrent reachability games where Player 1 can play a Joker to influence both the opponent and the next state, effectively modeling minimal cooperation. The authors formalize Joker strategies as cost-minimal strategies in the associated cost game and show key properties: Joker attractor strategies are cost-minimal, all winning Joker outcomes use the same number of Jokers, and Joker games are determinate; randomization is not required to win but can reduce the number of Jokers, and Joker distance strategies can further minimize moves. They establish deep connections between Joker strategies and admissible strategies, including conditions under which Joker-inspired, cost-minimal strategies are or are not admissible, and they extend the framework to randomized settings via probabilistic attractors and $pJAttr$, preserving cost-minimality. The practical impact is demonstrated through MBT test-case generation: Joker-inspired strategies yield test cases that reach goals with minimal cooperative input and outperform random testing in reachability and efficiency. The results provide a robust foundation for using semi-cooperative, Joker-based planning in both theoretical and applied settings, with potential extensions to multi-objectives and robustness against adversarial behavior.

Abstract

This paper coins the notion of Joker games, a variant of concurrent games where the players are not strictly adversarial. Instead, Player 1 can get help from Player 2 by playing a Joker move. We formalize these games as cost games and develop strategies that minimize the use of Jokers - viewed as costs - to secure a win with the least possible help. Our investigation studies the theoretical underpinnings of these games and their associated Joker strategies. In particular, when comparing our cost-minimal strategies with admissible strategies, we find out that they differ. Moreover, while randomization can be beneficial in conventional concurrent games, it does not aid in winning Joker games, although it can help reduce the number of needed Jokers. We also enhance our framework by introducing a secondary objective, namely by minimizing the number of moves executed by a Joker strategy. Finally, we demonstrate the practical advantages of our approach by applying it to test generation in model-based testing.

Figures (14)

• Figure 1: Concurrent game $G_{a \wedge b}$ with states $\{1,2,3,4,\smiley,\frownie\}$, Player 1 actions $\{a,b\}$, Player 2 actions $\{x,y\}$, and initial state 1. The $\textit{Moves}$ function and the enabling conditions $\Gamma_1$, and $\Gamma_2$ of $G_{a \wedge b}$ are represented by the edges. For example, the edge from state 1 to 2 indicates that $a \in \Gamma_1(1)$, $x \in \Gamma_2(1)$, and $2 \in \textit{Moves}(1,a,x)$.
• Figure 2: Illustration of the Joker attractor computation: it starts with states in R initially and then adds states using the attractor and Joker attractor operations, until a fixpoint is reached.
• Figure 3: Game $G_{a \wedge b}$, as in \ref{['fig:runningexample']}, but with dashed edges for moves of the Joker attractor strategy. On these dashed edges, if both the Player 1 and the Player 2 action are bold, then a Joker action is played. Otherwise, if only the Player 1 action is bold, then a (normal) Player 1 action is played. Note that non-dashed moves will never be played with this Joker strategy, no matter what Player 2 strategy or resolution of nondeterminism is used. Also, the dashed move from state 3 will not be played, because the strategy does not go there from the initial state.
• Figure 4: A cost-minimal strategy is depicted by the dashed edges. It plays a cost-0 $a$ action in state 1, and hopes Player 2 plays $x$ to arrive in with cost 0. If Player 2 plays $y$, the strategy plays a Joker in state 2 to win nevertheless. The computation of the Joker attractor yields that state 1 and 2 both have $J\textit{Rank}$ 1. Since initial state 1 is always reached first by the predecessor, the Joker attractor strategy (dotted) plays a Joker in state 1 immediately, and reaches directly from state 1. This example shows that cost-minimal strategies from state 1 may use fewer than $J\textit{Rank}(1)$ Jokers (against some but not all opponents), and that Joker attractor strategies from state 1 always use $J\textit{Rank}(1)$ Jokers.
• Figure 5: This game ${G}_{dist}$ has a cost-minimal strategy (dashed; cost 1 for the Joker used in state 4) that is not a Joker attractor strategy. For this game, there is a unique Joker attractor strategy (dotted). It selects Player 1 action $b$ from state 1 (to Joker state 3), since states 1 and 2 are added at the same iteration of the attractor over the set with Joker states 3 and 4. The witnessed attractor in iteration $k$ only selects Player 1 actions to states from an iteration $\neq k$. The dashed cost-minimal strategy requires fewer moves to reach than the dotted Joker strategy, namely 3 versus 4 moves.

Theorems & Definitions (68)

• Definition 3.1
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