- **Definition**: the study of strategically interdependent behavior. > [!info] > Strategic interdependence: what I do affects your outcomes and what you do effects my outcomes. ### The Basics simultaneous move games - e.g. soccer penalty kicks, prison interrogation, deciding whether to stop or drive at a stoplight Items that will be covered: 1. strict dominance 2. iterated elimination of strictly strategies 3. pure strategy Nash equilibrium 4. best responses 5. mixed strategy Nash equilibrium 6. weak dominance ### Extensive Form Games games whether players take turn moving - e.g. war, invasion plans, police searches Items that will be covered: 1. backward induction 2. subgame perfect equilibrium 3. crediable threats 4. trying hands 5. commitment problems 6. forward induction ### Advanced Strategic Form Games everything from chapter 1, except generalized - e.g. Does a striker kick left more frequently as his accuracy improves on the left side? Items that will be covered: 1. comparative statics 2. knife-edge equilibria 3. symmetric zero-sum games ### The prisoner's dilemma ![[Pasted image 20250407195656.png]] If player $y$ decides to Keep Quiet, the player $x$ is better to Confess. Also if the player $y$ Confess, the player $x$ is again better to Confess to minimize the punishment. Thus the strategy *Confess* **strictly dominates** strategy Keep Quiet. In general, strategy $x$ strictly dominates strategy $y$ for a player if $x$ generates a greater payoff than $y$ regardless of what the other player do. > [!tip] > Rational players never play strictly dominated strategies. ### Iterated Elimination of Strictly Dominated Strategies ![[Pasted image 20250407201036.png]] Suppose the following game. Player 1 should play Up, Middle or Down based on what Player 2 plays. However player 2 never should plays Right because his two other options are much better. ![[Pasted image 20250407201210.png]] Now Player 1 never should play Down because because Middle and Up are always better to play. ![[Pasted image 20250407201404.png]] Now Player 2 should always play Center because it always gives better result. ![[Pasted image 20250407201458.png]] And Player 1 should always play Middle because it is the best option to play. ![[Pasted image 20250407201538.png]] Thus the solution to this game is Middle-Center. ### Nash equilibrium A **Nash equilibrium** is a set of strategies one for each player, such that no player has incentive to change his or her strategy. > [!tip] > **Nash Theorem**: there must be at least one Nash equilibrium for all finite games. - We only care about individual deviations, not group deviations. - Nash equilibria are inherently stable: 1. What you are doing is optimal given what I am doing and vice versa 2. No regrets. **Key Takeaways about Nash Equilibrium:** 1. **It's about Stability:** It's a point where no one individually wants to change their current action. 2. **It's Self-Enforcing:** Once reached, it tends to stick because deviating alone makes you worse off. 3. **It's NOT Necessarily the Best Outcome:** In the Prisoner's Dilemma, both players would be better off if they both stayed silent (1 year each instead of 5). But that's not a stable equilibrium because each player has an individual incentive to confess, regardless of what the other does. 4. **Rationality Assumption:** It assumes players are rational and trying to maximize their own payoff, given their beliefs about what others will do. 5. **There can be Multiple Nash Equilibria:** The driving example has two (R,R and L,L). Some games have none (in terms of pure strategies). ![[Pasted image 20250407231421.png]] Both 1, 1 and 3, 3 are Nash equilibrium because in none of them one player can do better where the other one stays the same. Here is another formal definition of Nash equilibrium: ![[Pasted image 20250407232547.png]] #### Mixed Strategy Nash equilibrium Think back to Pure Strategy Nash Equilibrium (PSNE). That's a situation where every player has chosen one specific action, and nobody wants to change that single action, given what everyone else is doing. It's stable because everyone's choice is a best response to everyone else's specific choice. **When Pure Strategies Fail** Sometimes, like in Matching Pennies or Rock-Paper-Scissors, there is no such stable state using only pure strategies. Why? Because the essence of these games involves **outguessing** or **not being predictable**. - In Matching Pennies: If Player 1 always plays Heads, Player 2 will always play Tails to win. But if Player 2 always plays Tails, Player 1 will switch to Tails to win. But if Player 1 always plays Tails, Player 2 switches to Heads... you get into an endless cycle. There's no fixed pair of choices (H,H), (H,T), (T,H), or (T,T) where both players are happy sticking to their choice. - Predictability is punished. Knowing your opponent's move allows exploitation. **Introducing Randomness: The Mixed Strategy** Since being predictable is bad, the solution is to be unpredictable. But not just randomly unpredictable – strategically unpredictable. This is where mixed strategies come in. - A **mixed strategy** isn't choosing one action, but choosing a probability distribution over your available actions. - Instead of "I play Heads", it's "I will play Heads with probability p and Tails with probability (1-p)". **Expected Payoff: The New Goal** When randomness is involved, players can't guarantee a specific outcome. Instead, rational players aim to maximize their **expected payoff**. - **Expected Payoff** = (Payoff of Outcome 1 * Probability of Outcome 1) + (Payoff of Outcome 2 * Probability of Outcome 2) + ... and so on for all possible outcomes. It's like asking: "If this situation played out many times with these probabilities, what would my average payoff be?" Players choose the probabilities that give them the best possible average result, given the probabilities the other player is using. > [!tip] > - MSNE relies on the **indifference principle**: Each player chooses their probabilities to make the other player indifferent between the actions they are mixing.