Game Theory

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.

• 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

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.

• 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

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.

• 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

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.

• 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

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.

• 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

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.

• 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

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.

• 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