**1.**     **Intro to Game Theory** **1.1**  **Framework for Game Theory** ·       Be comfortable depicting games in both extensive form and normal form. What elements does each form convey? Order of moves? Player’s knowledge? Payoffs? **_Key Definitions:_** Know the distinctions between these terms. This will help you to make sure you give the right answer to the right question. ·       _Strategy_: A complete list specifying which move a player will make in every possible contingency, even if the player’s earlier move makes a later move unnecessary.  When specifying a player’s strategy, you need to specify one move for each distinct information set that a player has. So, a strategy can be specified as: “At info set #1, do X; At info set #2, do Y; etc.” ·       _Action_: a single move made by a player at a single node ·       _Outcome or Payoff_: the utility gain/loss that accrues to a player for a given sequence of moves by her and the other players. Sometimes we use money, years in prison, or years of life as a proxy for payoffs; this assumes risk neutrality in that metric. ·       _Equilibrium_: The combination of strategies by all players such that... (depends on the equilibrium concept). Remember - the equilibrium is the set of strategies, not payoffs ·       _Equilibrium Path of Play_: The sequence of moves we will see, given the equilibrium strategies of the players. If asked what the equilibrium outcome is, you can say what actions are taken on the equilibrium path. **1.2**  **Simultaneous Move Games** ·       Equilibrium Concept: Nash Equilibrium – The combination of strategies by all players such that none has a unilateral incentive to deviate: Everyone is playing a best response to the other player’s strategies. Be able to solve for pure strategy and mixed strategy Nash Eq. ·       Dominance - Know how to use iterated deletions of strictly dominated strategies to find an equilibrium. Be comfortable defining and identifying the following: o   Strictly dominant strategy o   Weakly dominant strategy o   Strictly dominated strategy o   Weakly dominated strategy ·       Toy Games - Recognize them and be able to use them for quick reference. What are the Nash Equilibria? Pure strategy or mixed? o   Prisoner’s Dilemma o   Battle of the sexes o   Stag hunt ·       Zero-sum games: players have strictly opposing interests. All equilibria have the same outcome (the “minmax”) and each player can guarantee her equilibrium outcome **1.3**  **Games of Incomplete Information** ·       How do economists and game theorists model uncertainty? Uncertainty about the opponents’ payoffs, but not about the possible moves or the order of moves. Games are based on some beliefs about the distribution of player types ·       Equilibrium Concept: Bayes-Nash Equilibrium: Each player’s strategy is a best response, maximizing expected payoffs, given available information. Players use Bayes’ rule to update their beliefs.  In equilibrium, beliefs are consistent with players strategies. (Note this method applies to all games of imperfect information) ·       Know how to convert games into the Bayesian normal form (process of transforming a game with incomplete information (where players have uncertain beliefs about other players' types or payoffs) into a normal form representation. In Bayesian normal form, each player's strategy is conditioned on their type and the game's payoffs are adjusted to reflect these conditional strategies and the players' beliefs. This allows for the analysis of the game using concepts like Bayes-Nash Equilibrium, which considers the strategies and beliefs of players in situations of uncertainty.) ·       Be comfortable solving for the critical probability that supports a particular equilibrium. **1.4**  **Dynamic Games** ·       Know how to draw games in extensive and normal form ·       Know how to define a subgame, singleton node, information set ·       Equilibrium Concepts: o   Subgame Perfect Nash Equilibrium (SPNE): Definition: NE within each subgame. Know how to find it, using backwards induction. SPE requires strategies to be optimal both on and off the equilibrium path. o   Nash Equilibrium:  Only requires a best response on the equilibrium path. SPNE refines NE (i.e. eliminates “unsatisfying” Nash Equilibria) by removing “non-credible” threats **1.5**  **Dynamic Games of Incomplete Information** ·       Multiple player types with some prior belief over the probability that a player is each type ·       Equilibrium concept: Perfect Bayesian Equilibrium: o   Adding in beliefs – with what probability do I think that I’m at a particular node in an information set? Use Bayes’ rule to update beliefs at each information set o   PBE consists of the following: §  At each information set, the player who moves assigns a probability to each node within the information set §  Given the players’ beliefs, everyone must be playing a sequentially rational strategy i.e. each player must be acting optimally at each information set given her beliefs and the other players’ subsequent strategies (the strategies that follow the information set). That is to say: strategies must now also be best responses to beliefs, in addition to best responses to other players’ strategies. §  Everyone’s beliefs along the equilibrium path are confirmed by the equilibrium play. (Or more formally, everyone uses Bayesian updating for their beliefs). §  [Beliefs off the equilibrium path are consistent with Bayes’ Rule – Strong PBE] ·       Example: Simple poker o   Know the setup of simple high low poker.  Know how to solve for equilibrium o   Be comfortable thinking about variations on the structure of the game. **1.6**  **Repeated Games** ·       Motivation: Why don’t we see people playing optimally in real life games ·       Includes stage games played multiple time with some discount rate on payoffs ·       Equilibrium Concept: Subgame Perfect Nash Equilibrium ·       Finite repeated games: solve using backwards induction ·       Infinitely repeated games: solve by calculating present value of total discounted payoff ·       Punishment Strategies – What condition does a successful punishment strategy have to satisfy? We need to make our opponent’s optimal strategy to be “Cooperate” instead of “Defect” o   “Tit for Tat” o   “Grim Trigger” **2.**     **Information Economics** **2.1**  **Adverse Selection (Readings: MWG pp. 437-450, lecture notes chapter 3)** ·       Basic Elements: o   One player or group (A) has information that the other (B) doesn’t o   The lack of information constrains the behavior of B, because B can’t distinguish between the different types of A (ability, quality, health, etc.) o   The resulting equilibria may be inefficient and reduce overall surplus (therefore “adverse”). Inefficiencies are usually due to beneficial transactions foregone, mismatch, or wasteful transactions. ·       Examples: Takeover game, lemons & Used Car Market, labor market selection **2.2**  **Signaling (Readings: MWG pp. 450-460, lecture notes chapter 4)** ·       Basic Elements o   One player or group (A) has information that the other (B) doesn’t o   Player(s) A take action to convey the information to player B, using a signal (beer/quiche, education) which is costlier for some types than others ·       The resulting equilibria may or may not be relatively more efficient than the non-signaling case, but there will usually be some winners and losers from signaling. Costly signals themselves introduce new inefficiencies (e.g. In Spence model, $ and pain spent on school) ·       PBE will be the equilibrium concept; we may end up with unsatisfying equilibria, based on unreasonable beliefs off the eqm path. **2.3**  **Signaling and the Spence Education Model** ·       Know how to draw the indifference curves, based on ability and education level ·       What do the IC and IR constraints look like (graphically, and algebraically)? ·       Solve for the range of possible separating equilibria ·       Solve for the existence of possible pooling equilibria ·       How do the intuitive criterion and the dominance refinement change our set of possible equilibria? **2.4**  **Screening (Readings: MWG pp. 460-466 & 488-501, lecture notes on screening and comments on competitive insurance model)** ·       Basic Elements: o   One player or group (A) has information that the other (B) doesn’t o   Player B takes action to induce A to reveal the hidden information o   The resulting equilibria may or may not be relatively more efficient than the non-screening case ·       Health Insurance & Competitive Screening (Rothchild-Stiglitz Model): o   Know how to draw the indifference curves, based on probability of illness o   What does a budget constraint look like for actuarially fair insurance? o   What do the IC and IR constraints look like? **2.5**  **Moral Hazard / Principal-Agent Problem (Readings: MWG pp. 478-488, lecture notes on principal-agent)** ·        Set up: Principal hires agent to work on her behalf, but agent’s actions are unobserved. What contract will provide the optimal set of incentives to the agent? ·       First Best: The contract that would be offered if effort was observable). ·       Standard P-A Model o   Effort Aversion o   Risk neutral Principal, risk averse Agent o   Effort is correlated with outcome, but not perfectly o   Higher effort is less enjoyable for worker o   CENTRAL TRADE-OFF: “Proper Incentives vs. Risk-Spreading” - Riskier contracts provide stronger incentives for higher effort, but require higher expected wages due to A’s risk aversion. ·       Know how to solve for the optimal contract.  In discrete cases this involves finding the optimal contract for inducing each level of effort and choosing the most profitable one for the principal **3.**     **Cooperative Game Theory** ·       How do we divide the profits to a partnership? ·       CORE: the set of allocations that is pareto optimal ·       Nash Bargaining o   Know how to solve for Nash bargaining solution (if feasible set is a triangle then the solution is the midpoint, if not maximize the product of utility) o   Only allocation that satisfies: §  Efficiency §  Invariance §  Symmetry §  Independence of Irrelevant Alternative ·       Shapley Value o   Know how to solve for the Shapley value allocation (calculating the marginal value of each player for each order of arrival) o   Only allocation that satisfies: §  Efficiency §  Symmetry §  Addition of games §  Zero player **4.**     **Social Choice & Voting** ·       The issue: How do we aggregate preferences across society as a whole, to enact an optimal policy? The ideal social choice rule captures the “true public preference” under any circumstances and does not provide incentive for strategic voting. ·       Definitions: o   Condorcet preferences / Condorcet cycle o   Borda count o   Pairwise majority voting ·       May’s Theorem – Majority voting is the only mechanism which satisfies these following axioms o   Symmetry among agents o   Neutrality among alternatives o   Positive responsiveness ·       Arrow Impossibility Theorem - What are the axioms? What are the results? What are the implications? o   Unrestricted domain for preferences o   Social welfare functional is rational (complete and transitive) o   Paretian o   Pairwise independence (IIA) o   Non-dictatorial