Source: Levelica Canonical: https://www.levelica.com/en/guides/game-theory # Game Theory ## Summary Modeling interactions that hinge on others' choices, then solving for equilibrium to predict the outcome. ## Description Game theory turns situations where several people shape each other's outcomes into formal models you can solve. You fix who plays, what options each one has, and what payoff every combination of choices produces. Then you look for the state where nobody gains by switching alone. This guide covers moving a situation with an opponent into a model and judging it by equilibrium. It doesn't cover setting direction by matching goals to resources, or correcting the biases in your own judgment. --- ## Level Overview | Level | Proficiency Stage | Hashtags | | ----- | ------------------------------- | ----------------------------------------------------------------------------------------- | | Lv.1 | Game Identification | #StrategicInteraction #PlayersStrategiesPayoffs #PayoffMatrix #PrisonersDilemma | | Lv.2 | Static Game Solving | #DominantStrategy #BestResponse #NashEquilibrium #IteratedElimination | | Lv.3 | Full Equilibrium Search | #MixedStrategy #Rationalizability #CoordinationGames #EquilibriumMultiplicity | | Lv.4 | Dynamic Game Analysis | #ExtensiveForm #BackwardInduction #SubgamePerfection #CredibleCommitment #BargainingGames | | Lv.5 | Incomplete Information Handling | #RepeatedGames #FolkTheorem #BayesianGames #SignalingGames #ReputationEffects | | Lv.6 | Rule Design | #MechanismDesign #AuctionDesign #MatchingMarkets #IncentiveCompatibility #GlobalGames | | Lv.7 | Solution Concept Extension | #SolutionConceptDesign #EquilibriumRefinement #GameTheoryPedagogy #StrategicModelingCanon | > Proficiency mapping source: The stage order follows the course sequence of MIT OpenCourseWare 14.126 (solution concepts for static games, then extensive-form games, then repeated games and advanced topics) together with the learning outcomes of the Warwick game theory module. The stage names come from the core behavior of each level. Game theory has no internationally recognized grading system, so no rating codes are used as stage names. > **Integrative Framework:** The seven levels of this guide are not a transcription of one curriculum. They split where four independent axes meet. The course sequence of MIT OpenCourseWare 14.126 carries the **game structure axis**, the class of game widening from simultaneous choice to sequential choice and then to repetition and incomplete information. The solution concept account in the Stanford Encyclopedia of Philosophy and the Warwick module learning outcomes carry the **solution concept axis**, the judging tool sharpening from eliminating dominated strategies to best response and Nash equilibrium, then to refinements of equilibrium. Fixing the class of game still leaves the choice of solution concept open, so the two axes do not overlap. Two axes specific to this guide join them. The **modeling role axis** runs from the seat where you read models other people built, to the seat where you translate your own matter into a model, to the seat where you write rules and widen the language of analysis. The schema induction work of Gick & Holyoak(1983), which found that transfer needs a shared structure pulled from different cases, backs the transfer mechanism of this axis. The **prediction testing axis** runs from stopping once the equilibrium is computed to checking it against real results and folding the finite reasoning depth of the other side into the model. Shapiro et al.(2014), who split reasoning depth into tiers, supply the observable measure. Level boundaries sit where all four axes step up at once. > **Cross-Validation:** The level boundaries were set by checking whether separate sources converge on the same point. > > - **Lv.2 to Lv.3 boundary**: The Warwick module lists strategy, best response, and equilibrium side by side as learning outcomes, and the Stanford Encyclopedia of Philosophy defines Nash equilibrium as a state where no one can improve their payoff by changing only their own strategy. The MIT sequence also places rationalizability and Nash equilibrium in one block. Three sources point at the same bundle, so the split here falls between solving problems someone else set and carrying your own matter into a model and on to equilibrium. > - **Lv.3 to Lv.4 boundary**: MIT 14.126 places extensive-form games, backward induction, and subgame perfection only after the solution concepts for static games are finished. The Stanford Encyclopedia of Philosophy likewise defines a subgame perfect strategy separately, after Nash equilibrium, as one that picks the highest-payoff path of the relevant subgame at every node reached. Two sources put a step in the same place independently, and the game structure axis and the solution concept axis each climb a rung here. > - **Lv.5 to Lv.6 boundary**: The last block of the MIT sequence runs through repeated games, reputation, global games, and cooperative solutions, marking where analysis hands off to design. The Warwick module learning outcomes stop before that. The point where the modeling role axis turns from using models to writing rules overlaps with the gap in scope between the two sources. > - **Lv.6 to Lv.7 boundary**: Above this, no curriculum names a next step. Both the MIT course sequence and the Warwick learning outcomes stop at the highest reachable point, so Lv.7 is defined not by a rating but by how long the solution concepts you leave behind keep getting referenced. --- ## Level 1 #StrategicInteraction #PlayersStrategiesPayoffs #PayoffMatrix #PrisonersDilemma ### Summary Spotting interactive situations and naming the three parts of a game. ### Description You have just stepped into game theory. You separate situations your choice alone settles from situations the other side's choice helps settle. In a given example you point to the players, the options each one holds, and the payoff attached to every combination. You read a table where two people each pick between two options. You don't build tables yet. You read models other people made and get used to the shape of an interaction. ### Checklist - [ ] Can tell apart, with examples, situations where the outcome depends on the other side's choice and situations where it doesn't - [ ] Can list every player and every option each one holds in a given example - [ ] Can point out, in a 2x2 table, which player each number in a cell pays out to - [ ] Can assume the other side's choice is fixed and point to the option that pays you most - [ ] Can explain why individual gain and group gain split apart in the Prisoner's Dilemma - [ ] Can pick one situation you ran into recently and write the other side, the options, and the results into a table > If you've checked off most of this list, you're ready to enter Lv.2 Static Game Solving, where you turn real situations into tables yourself. Gick & Holyoak(1983) report that one case alone yields no structure, and that a shared structure sticks as a schema only when you compare cases that look different. Say the players, options, and payoffs of one situation out loud, then add a case from another field in the same 2x2 frame. What you left out shows up beside what they share. --- ## Level 2 #DominantStrategy #BestResponse #NashEquilibrium #IteratedElimination ### Summary Building payoff matrices yourself to find pure-strategy equilibria. ### Description You start moving real interactions into tables on your own. You set the players and each one's strategy set, fill the payoffs in as numbers, and build a payoff matrix. You look for a dominant strategy that does better no matter what the other side picks, then delete dominated strategies one by one to narrow the field. You mark each player's best response on the table and point to the cell where they meet, the pure-strategy Nash equilibrium. ### Checklist - [ ] Can move one real situation into a two-player payoff matrix and write it out as a table - [ ] Can tell strictly dominant strategies from weakly dominant ones and find both in a table - [ ] Can delete dominated strategies one by one and derive what survives - [ ] Can mark each player's best response on the table and point to the cells where they meet - [ ] Can find a pure-strategy Nash equilibrium and confirm that nobody gains by switching alone - [ ] Can give an example of a game with a pure-strategy equilibrium and one without, and tell them apart - [ ] Can walk someone else through a payoff matrix you built and give reasons for the numbers you set > If you've checked off most of this list, you're ready to enter Lv.3 Full Equilibrium Search and work out mixed strategies and multiple equilibria on your own. The Stanford Encyclopedia of Philosophy defines Nash equilibrium as a state where no one can improve their payoff by changing only their own strategy. Turn that definition into a test and hunt for counterexamples to your own answers. That is what moves equilibrium from something you memorized to a tool in your hands. --- ## Level 3 #MixedStrategy #Rationalizability #CoordinationGames #EquilibriumMultiplicity ### Summary Computing mixed strategies and multiple equilibria on your own. ### Description This is where you cross from leaning on others to working alone. In games with no pure-strategy equilibrium you compute the probabilities of a mixed strategy equilibrium and check them against the condition that leaves the other side indifferent. In coordination games with several equilibria you predict which one gets reached and give reasons. The problems are no longer someone else's. You take a case you ran into, model it alone, and carry it to a conclusion. ### Checklist - [ ] Can compute the probabilities of a mixed strategy equilibrium in games with no pure-strategy equilibrium - [ ] Can derive the equilibrium probabilities from the condition that leaves the other side indifferent, then check the arithmetic - [ ] Can predict which of two or more equilibria a coordination game reaches and give reasons - [ ] Can separate the strategies that survive the rationalizability test from the ones that drop out - [ ] Can set up the equilibrium conditions for games with three or more players or with a continuum of options - [ ] Can model a case you ran into yourself and produce a one-page document with the conclusion and the reasoning - [ ] Can build two models of the same situation that differ only in the payoffs and explain why the conclusions differ - [ ] Can point to one assumption in your model that would flip the conclusion and check whether it holds in reality > If you've checked off most of this list, you're ready to enter Lv.4 Dynamic Game Analysis, where the order of moves changes the result. Gick & Holyoak(1983) report that transfer to an unseen third case needs a shared structure pulled from comparing two. Model the same situation twice, once as simultaneous choice and once as sequential, mark where the answers split, then run that same contrast on a case you have never seen before. That pulls the shift forward. --- ## Level 4 #ExtensiveForm #BackwardInduction #SubgamePerfection #CredibleCommitment #BargainingGames ### Summary Solving game trees and telling credible threats from empty ones. ### Description You move from applying rules to adapting to the situation. You draw sequential interactions as an extensive-form tree and solve it by backward induction, working up from the last node. Subgame perfect equilibrium, which picks the best line of the subgame starting at every node you reach, filters out threats that would cost the threatener to carry out. You handle bargaining order and each side's share with the same tool. ### Checklist - [ ] Can draw a sequential interaction as an extensive-form tree and mark who moves at each node - [ ] Can solve a tree by backward induction and mark the equilibrium path - [ ] Can find a threat that would cost the threatener to carry out and rule it not credible - [ ] Can compare two trees to explain how the result changes when the order of moves changes - [ ] Can design a way to make your own threat credible, such as an upfront commitment or cutting off your retreat - [ ] Can model a bargaining round of offers and counteroffers and compute the share going to whoever offers first - [ ] Can apply a model to a real negotiation or auction and record the gap between the prediction and the result - [ ] Can estimate how many moves ahead the other side weighs best responses and adjust your own choice > If you've checked off most of this list, you're ready to enter Lv.5 Incomplete Information Handling and take on repeated relationships and hidden information. Shapiro et al.(2014) split the depth of strategic reasoning into level-k tiers, making how far the other side follows observable. Write your estimate of their depth down, compare it against what happened, then repeat the game and move conditions until the conclusion flips. That threshold shows why the next tool exists. --- ## Level 5 #RepeatedGames #FolkTheorem #BayesianGames #SignalingGames #ReputationEffects ### Summary Handling repetition and hidden information to steer team decisions. ### Description You go past single play into repeated relationships and hidden information. You work out, from how heavily future payoffs count, the condition that keeps cooperation alive, and find the range where punishing a defection holds as an equilibrium. When the other side's type is unknown, you set it up as a Bayesian game and update your beliefs, and read signaling games where a costly signal separates the types. Your reach widens from personal analysis to team decisions. ### Checklist - [ ] Can compute the discount factor condition that keeps cooperation alive in a repeated game - [ ] Can judge whether a punishment strategy against defection holds up as an equilibrium - [ ] Can set up a situation with an unknown opponent type as a Bayesian game and state the prior probabilities - [ ] Can update your belief about the other side's type after new information and recompute the best response - [ ] Can tell a separating equilibrium from a pooling one when the signal is costly - [ ] Can leave a record of your model analysis being adopted in a real team or organizational decision - [ ] Can coach three or more people step by step, from writing a payoff matrix to deriving the equilibrium - [ ] Can point to situations where a game model clouds the judgment instead and pick a different approach > If you've checked off most of this list, you're ready to enter Lv.6 Rule Design, writing rules that make the outcome you want an equilibrium. MIT OpenCourseWare 14.126 puts repeated games, reputation, and cooperative solutions in its final block, which shows the order in which analysis hands off to design. Start by fixing the conclusion first and writing backward to the rules that turn it into an equilibrium. That is how the design instinct forms. --- ## Level 6 #MechanismDesign #AuctionDesign #MatchingMarkets #IncentiveCompatibility #GlobalGames ### Summary Designing rules and institutions that steer how participants behave. ### Description You move from the seat where you solve games to the seat where you build them. You design rules and reward structures backward so the outcome you want comes out as an equilibrium, and you secure incentive compatibility so telling the truth pays the participant best. You compare auction formats and allocation rules to pick the one that fits the situation, and you use a model to explain the threshold where a crowd guessing at each other's judgment tips one way. ### Checklist - [ ] Can design rules and reward structures backward so the outcome you want becomes an equilibrium - [ ] Can design an incentive structure that pays off for telling the truth and verify the conditions it holds under - [ ] Can compare three or more auction formats or allocation rules and pick the one that fits the situation - [ ] Can run a rule you designed in the field and measure whether participant behavior matched the prediction - [ ] Can explain the threshold condition where a crowd guessing at each other's judgment tips one way - [ ] Can publish your analysis method as a lecture or article and get other organizations to adopt it - [ ] Can find three or more routes that get around the intent of a rule you designed and block them in advance > If you've checked off most of this list, you're ready to enter Lv.7 Solution Concept Extension and fill the gaps that existing equilibrium concepts miss. Both the MIT course sequence and the Warwick module learning outcomes hit their ceiling here. Collect the cases where your prediction missed and write down which assumption broke in each one. Once nobody has laid out the order for you, that is where your own questions come from. --- ## Level 7 #SolutionConceptDesign #EquilibriumRefinement #GameTheoryPedagogy #StrategicModelingCanon ### Summary Creating new solution concepts that change how the field thinks. ### Description You change the method of analysis itself. You point to situations existing equilibrium concepts cannot explain and put forward a solution concept or model class that fills the gap. The frame you proposed travels into research and practice elsewhere, and people you trained go independent and publish models under their own names. No curriculum and no rating names the next step, so the only proof of arrival is how long the concepts you leave behind keep getting cited. ### Checklist - [ ] Can point to situations existing equilibrium concepts cannot explain and put forward an alternative - [ ] Can confirm that the solution concept or model class you proposed is in use in three or more other fields - [ ] Can show that your research or method has been cited continuously for five years or more - [ ] Can confirm that five or more people you trained have gone independent and published their own models - [ ] Can confirm that your concept has been folded into a standard textbook or a university curriculum - [ ] Can state the conditions under which your solution concept fails and publish them so later generations can test it > If you've checked off most of this list, you're standing where the language for handling situations with an opponent gets rewritten. No curriculum and no rating names anything above this, so what you reached is measured not by how many papers you published but by which concepts the next generation takes for granted. --- ## References - [MIT OpenCourseWare 14.126 — Game Theory (Syllabus)](https://ocw.mit.edu/courses/14-126-game-theory-spring-2016/pages/syllabus/) — MIT OpenCourseWare - [EC941 Game Theory — Module Learning Outcomes](https://warwick.ac.uk/fac/soc/economics/current/modules/ec941/) — University of Warwick, Department of Economics - [Level-k Reasoning in a Generalized Beauty Contest](https://doi.org/10.1016/j.geb.2014.04.002) — Shapiro, Shi & Zillante (Games and Economic Behavior) - [Game Theory (Stanford Encyclopedia of Philosophy)](https://plato.stanford.edu/entries/game-theory/) — Stanford Encyclopedia of Philosophy - [Schema induction and analogical transfer](https://doi.org/10.1016/0010-0285(83)90002-6) — Gick & Holyoak (Cognitive Psychology)