Game theory - Classification of Games

Types of Games in Game Theory Introduction Game theory classifies games along several important dimensions, each capturing different strategic features. Understanding these classifications helps us recognize which solution concepts and analytical tools apply to a particular problem. Think of these dimensions as a taxonomy—they help us organize games the way a biologist organizes living organisms. No single classification tells the whole story; instead, a real game is typically described using multiple categories. Cooperative vs. Non-Cooperative Games The most fundamental distinction separates games by whether players can make binding agreements. Non-cooperative games assume that players cannot form enforceable agreements. Any deals between players must be self-enforcing—meaning each player follows the agreement only if it's in their individual interest to do so, backed by credible threats. For example, in a business negotiation without a legal contract, you follow through only if breaking your word would be worse than honoring it. Most of the games you'll encounter in an introductory course are non-cooperative. Cooperative games allow players to form binding, externally enforced agreements. Think of a formal contract: both parties are legally compelled to follow it, regardless of individual incentives. In cooperative games, the analysis shifts from predicting individual choices to determining how players should divide the joint gains from cooperation. Rather than asking "what will each player do?", we ask "how should the collective payoff be distributed?" The key practical difference: in non-cooperative games, we analyze behavior through concepts like Nash equilibrium. In cooperative games, we focus on coalitions and fair allocation schemes (like the core or Shapley value). Symmetric vs. Asymmetric Games This classification concerns whether all players face identical strategic situations. In a symmetric game, every player receives the same payoff when each chooses the same strategy. Consider a simple matching game where both players choose "heads" or "tails": if both choose the same option, each gets payoff 1; if they differ, each gets 0. The payoff structure treats both players identically. In an asymmetric game, players have either different strategy sets or different payoffs for the same strategy combination. The classic example is the ultimatum game: one player (the proposer) divides a sum of money, and the other player (the responder) can accept or reject it. These roles are fundamentally different, so the players face asymmetric situations. Real-world negotiations, auctions with different bidders, and employment relationships are typically asymmetric. Why does this matter? Symmetric games often have elegant solutions, and we can sometimes assume that all players use the same strategy in equilibrium. Asymmetric games require more careful analysis because players' optimal responses differ. Zero-Sum vs. Non-Zero-Sum Games This dimension describes whether total payoffs are constant or variable. Zero-sum games have one essential property: the sum of all players' payoffs is always zero (or some constant). One player's gain is exactly another player's loss—there's no mutual benefit, only redistribution. Classic examples include: Poker: the chips won by one player come from others Matching pennies: if one player wins $1, the other loses $1 Chess (if we assign wins as +1 and losses as -1) Non-zero-sum games (also called constant-sum games when the total is a different constant) allow outcomes where all players simultaneously gain or lose. The prisoner's dilemma is the canonical example: both players could gain if they cooperated, but individual incentives pull them toward mutual loss. Most economic and social interactions are non-zero-sum because trade, innovation, and cooperation can make everyone better off. The strategic implications are profound. In zero-sum games, my gain directly hurts you, so we're in pure conflict. In non-zero-sum games, we often face a mixture of cooperation and competition—both are possible. Simultaneous vs. Sequential Games These categories describe the timing of decisions. In simultaneous games, all players choose their actions at the same time without knowing what the others are choosing. Each player must commit to a strategy without observing others' choices. Rock-paper-scissors, sealed-bid auctions, and most strategic business decisions fit this pattern. We typically represent simultaneous games using a normal form (a table or matrix showing payoffs for each combination of strategies). In sequential games, players move one after another, with earlier actions becoming known before later players decide. Chess is the clearest example: you see your opponent's move before making yours. We represent sequential games using an extensive form (a game tree showing the sequence of moves and information). Why does this matter? Sequential games often have advantages for the player who moves first (first-mover advantage), and players can condition their strategies on observed history. Simultaneous games require each player to guess what others will do. Perfect vs. Imperfect Information This classification describes what players know about the past. A perfect information game gives every player complete knowledge of all actions taken previously in the game. Chess and tic-tac-toe are perfect information games—the entire history is visible on the board. When it's your turn, you know exactly what happened before. An imperfect information game hides some previous actions from some players. In poker, you don't see your opponents' cards even after they've been dealt. In business negotiations, you may not know what offers your competitor has received. Even in chess, if moves were made in secret (you only see your opponent's last move, not their previous ones), it would be imperfect information—though that's an unusual variant. Imperfect information creates strategic complications: players must form beliefs about hidden information, and these beliefs affect optimal play. This directly connects to our next classification. Complete vs. Incomplete Information This describes what players know about payoffs and types. Complete information means every player knows all other players' strategy sets and payoffs. You understand exactly what outcomes are possible and what each outcome is worth to everyone. This is the simplest assumption and underlies basic Nash equilibrium analysis. Incomplete information occurs when players are uncertain about others' payoffs, preferences, or available strategies. You might not know if your negotiating partner truly values the good at $100 or $200. You might not know if your competitor has a cost advantage. In card games, you don't know what cards others hold, which translates to uncertainty about their available moves and effective payoffs. Incomplete information is modeled using Bayesian games: we assign each player a type (representing their private information) drawn from some probability distribution known to all players. Each type has different payoffs. Players have beliefs (probability distributions) about what types other players might be. The key insight: even though the underlying game has some uncertainty, assuming common knowledge of probability distributions allows us to analyze behavior through expected payoffs. This connects uncertainty about payoffs to the statistical reasoning that everyone can share. Extensive-Form Games and Information Sets When we represent sequential games using a game tree, we must capture not just the sequence of moves, but also what players know at each decision point. An information set is a node in the game tree where a player cannot distinguish between different histories. If you cannot tell which of several situations you're in, they belong to the same information set. For example, in poker before the flop is revealed, you know your own cards but not opponents' cards—you cannot distinguish between worlds where your opponents hold different cards. These worlds are in the same information set for you. A game has perfect information if every information set contains a single node—each player always knows exactly where they are in the game tree. A game has imperfect information if some information sets contain multiple nodes—a player faces genuine uncertainty about the game state. This diagram shows a game tree with information sets (the dotted lines connecting nodes). A player cannot distinguish between nodes in the same information set. Other Game Classifications Stochastic Games and Games with Chance Many real-world situations involve randomness. Stochastic outcome games contain chance moves—random events that affect payoffs. We model this by introducing a fictitious "nature" player that makes random moves according to specified probability distributions. For instance, in poker, nature deals the cards according to random probability. Players typically maximize expected payoff (the probability-weighted average) when chance is involved. This differs from minimax reasoning (which protects against worst-case outcomes), though sometimes a mix of both applies. Repeated Games Repeated games involve the same stage game being played multiple times. If the game repeats finitely many times, players' behavior may differ significantly from a one-shot game because they don't have infinite future to build reputation. If the game repeats infinitely, the folk theorem shows that an enormous range of equilibrium outcomes becomes possible—far more than in a single-shot game. This is because threats of future punishment become credible. <extrainfo> Combinatorial Games Combinatorial games like chess, Go, and Hex are characterized by a large, often astronomically large, number of possible move sequences. These games are typically perfect information games without chance moves, but their complexity makes them difficult to solve by exhaustive search. Advanced techniques like alpha-beta pruning and neural network evaluation are used in practice. Discrete vs. Continuous Games In discrete games, strategies must be chosen from a finite or countably infinite set—for instance, choosing an integer number of units to produce, or selecting one of several locations. In continuous games, strategies are chosen from a continuous set (like any non-negative real number). A firm in Cournot competition can produce any non-negative quantity, making this a continuous game. Continuous games often allow elegant mathematical solutions using calculus, but require strategies to be differentiable functions. This diagram shows the reaction curves in a continuous Cournot game, where quantities are chosen from continuous ranges. Evolutionary Games Evolutionary game theory studies populations whose strategies change over time according to dynamics like imitation, trial-and-error learning, or natural selection. Rather than assuming rational players solving optimization problems, we model how strategies spread or decline based on their relative success. The concept of an evolutionarily stable strategy (ESS) describes a population composition that resists invasion by mutant strategies. Population dynamics are often modeled as Markov chains that eventually converge to an ESS. Metagames Metagames are games about games—they relate closely to mechanism design theory, which studies how to design rules that produce desired outcomes when players behave strategically. </extrainfo> Key Takeaway Games are classified along multiple independent dimensions: whether binding agreements are possible, whether all players face identical or different situations, whether the total payoff is zero or variable, the timing and information structure of decisions, and various other features. A complete description of a game typically invokes several of these categories. This classification system helps us select appropriate solution concepts and predict which strategic considerations matter most.