8. Game Theory

The grammar of strategic choice — from normal-form games and best responses through Nash equilibrium, mixed strategies, backward induction in dynamic games, and repeated games.

• Normal-Form Games and Best Responses — Look directly at the two-player payoff table. Each of the four cells lists both players' payoffs. In each column, mark the first player's best answer. In each row, mark the second player's best answer. The cell where both marks land gets a star. That cell is a pure-strategy Nash equilibrium. Neither player wants to move away alone. For example, say you and a 🧑 friend each pick heads or tails on a coin. If your friend picks heads, one answer is best for you. Your pick likewise settles your friend's best answer. The pairing where both answers point at each other is the equilibrium.
• Mixed Strategies and the Indifference Condition — Watch the second player mix two strategies by probability. Slide the mixing probability back and forth. As the rival's mix changes, my expected payoff from each strategy moves along a straight line. At the mix where the two lines cross, neither strategy is better. Only that indifferent mix can appear in a mixed equilibrium. Making your rival indifferent is what pins down your mix. For example, say a 🧑 friend takes a penalty kick with a ⚽ ball against a goalkeeper friend. Always kicking left lets the keeper guard only left. So the kicker mixes left and right kicks. The mix makes the keeper equally unhappy guarding either side.
• Zero-Sum Games and Minimax — Look at a game where one player's gain is exactly the other's loss. Edit the payoff numbers yourself. I try to make my worst case as good as possible. My rival tries to make my gains as small as possible. When the two calculations agree, a saddle point appears and the equilibrium is pure. When they disagree, both players must mix, and the game gets a value as in the formula below. For example, bet one 🍬 candy on guessing heads or tails of a coin. Guess right and the candy is yours. Guess wrong and your 🧑 friend takes it. Whatever you win, your friend loses. So both of you mix to stay unreadable.
• Nash Equilibrium and Best-Response Correspondences — Draw both players' best-response curves on one plane. The blue curve and the orange curve appear together. The blue curve is my best answer to the rival's mixing probability. The orange curve is the best answer in the other direction. Where they cross, each is a best response to the other. That crossing is a Nash equilibrium. Pure equilibria sit at the corners, and the mixed one is starred inside. For example, say you and a 🧑 friend each choose left or right. Your best answer changes with your friend's choice. Your friend's best answer changes with yours too. The pairing where both answers fit each other is the equilibrium.
• Iterated Elimination of Strictly Dominated Strategies — Watch dominated strategies get deleted one by one. Push the round slider to eliminate them in order. If a strategy always does worse than another one, no rational player uses it. Delete it and the table shrinks. In the smaller table, another bad strategy shows up. In this scene C, M, and D vanish in turn until only U remains. For example, say a 🏪 shop sells three kinds of 🍬 candy. One kind is always smaller and less tasty than another kind. Nobody ever picks it, so it can leave the shelf. Once it is gone, another always-worse candy stands out. Deleting step by step leaves the final choice.
• Extensive-Form Games and Backward Induction — Draw the entry game as a tree. The order of moves is visible at a glance. Solve it by reading backwards from the end. First fix the last mover's best answer. Knowing that answer settles the earlier player's choice too. A threat to fight is not credible, because fighting hurts the threatener once entry happens. So that branch gets pruned away. For example, say a neighborhood has one snack 🏪 shop and a new one wants to open. The old owner threatens a price war. But once the new shop opens, fighting costs the old owner too. The newcomer knows this and enters anyway.
• Repeated Games and Cooperation — Watch the Prisoner's Dilemma repeated forever. The strategy punishes any betrayal forever. Betray today and you gain a lot right now. But from tomorrow your partner betrays too, and the losses keep coming. How much you care about tomorrow decides the choice. Care enough about the future, and cooperation pays and holds. The boundary value comes from the formula below. For example, say you and a 🧑 friend take turns lending a ⚽ ball every day. Hog the ball today and today is fun. But from tomorrow your friend never lends it to you. If you want to keep playing together, keeping today's promise wins.