Cartesian Game Notation: A Pedagogic Geometry of Strategic Outcomes

Scholarly abstract This paper establishes a one-to-one correspondence between the four outcome cells of symmetric 2 × 2 games and the four quadrants of the Cartesian plane. Building on the classic R, S, T, P labels, it introduces Caprazli notation, which adds payoff-sign superscripts (R++, S-+, T+-, P--) so that each outcome’s quadrant is determined by who gains and who loses relative to a reference point. In this coordinate system, the horizontal axis is Player 1’s payoff, the vertical axis is Player 2’s payoff, and the standard quadrant ordering becomes a map of joint gain, unilateral exploitation, mutual loss, and temptation. The framework classifies all twelve canonical symmetric 2 × 2 games by the quadrant locations of their Nash equilibria and formulates the Wobble Principle, which describes how sign changes in payoffs trigger regime shifts between game types. This geometric representation yields a single “payoff quadrilateral” that links normal-form games to continuous payoff-space analysis and supports visual reasoning about game transformations in both research and teaching. Plain-language summary Many real situations, such as whether to keep a promise or free-ride on others, can be modeled as simple two-choice games with four possible outcomes. Traditionally these outcomes are shown in a table and labeled with letters such as Reward, Sucker, Temptation, and Punishment, which can feel abstract. This paper replaces the table with a graph: each outcome is placed in one of four quadrants, depending on whether each player gains or loses compared with a baseline. Outcomes where both gain sit in the upper-right quadrant, both lose in the lower-left, and the two mixed outcomes occupy the remaining quadrants, with plus and minus signs in the labels making this explicit. As incentives change, outcomes move between quadrants, and the underlying game changes character, turning cooperation problems into coordination or conflict and back again. This geometric picture lets students see at a glance what kind of game they face and how small shifts in payoffs can fundamentally change strategic behavior. Significance statement Despite the obvious “four cells, four quadrants” parallel, game-theory pedagogy has lacked a standard mapping from symmetric 2 × 2 games into Cartesian space. Caprazli notation and the associated quadrant framework close this gap by tying each outcome’s label directly to the sign pattern of players’ gains and losses and by organizing all twelve canonical symmetric games in a single geometric atlas. This makes regime shifts between game types literally visible as movements across quadrant boundaries and offers a compact way to teach comparative statics and incentive changes that is difficult to achieve with payoff tables alone. Because the construction is lightweight and compatible with standard R, S, T, P labels, it can be adopted in introductory courses while also connecting to more advanced payoff-space and dynamical analyses, giving educators and researchers a common visual language for strategic interaction. Technical summary We introduce a systematic mapping between the four outcome cells of symmetric 2 × 2 games and the four quadrants of the Cartesian plane. The standard game-theoretic notation—Reward (R), Sucker (S), Temptation (T), Punishment (P)—is augmented with payoff-sign indicators, which we term Caprazli notation: R++ (Quadrant I), S-+ (Quadrant II), P-- (Quadrant III), T+- (Quadrant IV). This representation makes the geometric structure of strategic interaction immediately visible: the horizontal axis encodes Player 1’s payoff sign, the vertical axis encodes Player 2’s payoff sign, and the counter-clockwise quadrant labelling follows standard mathematical convention. We classify all twelve symmetric 2 × 2 games by the quadrant locations of their equilibria and introduce the Wobble Principle—the observation that payoff-sign changes induce regime transitions between game types. The framework offers pedagogical advantages, enables visual reasoning about game transformations, and connects discrete game theory to continuous payoff-space analysis.