“Life must be understood backward, but … it must be lived forward.” Game Theory “Life must be understood backward, but … it must be lived forward.” - Soren Kierkegaard James Bednar

Game Theory: “When you think carefully before you act--when you are aware of your objectives or preferences and of any limitations or constraints on your actions, and choose your actions in a calculated way to do the best thing according to your own criteria--you are said to be behaving rationally. Game theory adds another dimension to rational behavior, namely interaction with other equally rational decision makers. In other words, game theory is the science of rational behavior in interactive situations.” Dixit and Skeath, Games of Strategy

Definition of a game Must consider the strategic environment Who are the decision makers (PLAYERS)? What are the feasible ACTIONS? What are the outcomes (PAYOFFS)? Rules of the game What is the nature of the conflict? Zero-sum or No zero-sum What is the nature of interaction? One time or Iterated What information is available? Simultaneous or Sequential

Simultaneous or Sequential? Players must move without knowledge of what their rivals have chosen to do. Try to figure out what opponent is going to do right now; opponent is trying to do the same, and recognizes this, and recognizes that you recognize this..... Battle of the Bismarck Sea Sequential There is a strict order of play, and players know what players who have gone before them have done. Each player must consider: if I make this move, how will my opponent respond? Senate Race Game, Tic-Tac-Toe

Player 1 You have $100 that you must share between yourself and player 2. You must make an offer as to how to split the $100. Player 2 must either reject or accept the offer. If player 2 rejects the offer, neither of you will get anything. If player 2 accepts the offer, you divide the money accordingly. What is your offer?

Player 2 Do you accept or reject the offer?

Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

You have $100,000 that you must divide between yourself and another person (player 2) in $1,000 increments. You must make an offer as to how to divide the money. If player 2 accepts the proposal, you divide the money accordingly. If player 2 rejects the proposal, he or she gets $80,000. What do you offer?

Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

Decision Trees Sequential games can be represented with decision trees: 0,2 t B n A b 2,1 s 1,0 Assume that the higher the number, the better.

Assume that the higher the number, the better. 0,2 t B n A b 2,1 s 1,0

Assume that the higher the number, the better. 0,2 t B n A b 2,1 s 1,0

Solving Sequential Games Start with the last move in the game Determine what that player will do Trim the tree This results in a simpler game Repeat the procedure, if necessary

David Hume   “Your corn is ripe to-day; mine will be so to-morrow. ‘Tis profitable for us both, that I should labour with you to-day and that you should aid me to-morrow. I have no kindness for you, and know you have as little for me. I will not, therefore, take any pains upon your account; and should I labor with you upon my own account, in expectation of a return, I know I shou’d be disappointed, and that I shou’d in vain depend upon your gratitude. Here I leave you to labour alone: You treat me in the same manner. The seasons change; and both of us lose our harvests for want of mutual security.” Treatise on Human Nature

Should I help my neighbor? You have to decide first. We’ll call you “A”. You’re trying to decide between “help” and “don’t help” A h ~h

Should I help my neighbor? Your neighbor goes second. Call him “B”. He has to decide whether to return the favor. B r ~r A h ~h Now have to add payoffs to the end nodes.

Should I help my neighbor? Assume that the higher the number, the better. 2,1 B r ~r h A 0,2 ~h 1,0 A = You. B = Your Neighbor. h = Help B with crops. r = Return the favor to A.

Solving Sequential Games Start with the last move in the game Determine what that player will do Trim the tree This results in a simpler game Repeat the procedure, if necessary

Solving with Rollback Assume that the higher the number, the better. 2,1 r B h A ~r 0,2 ~h 1,0

Solving with Rollback Assume that the higher the number, the better. 2,1 A B h ~h r ~r 0,2 1,0

What would happen if I didn’t use rollback?

Reasoning Forward Assume that the higher the number, the better. 2,1 B 0,2 ~h 1,0

Reasoning Forward Assume that the higher the number, the better. 2,1 B 0,2 ~h 1,0

Reasoning Forward Assume that the higher the number, the better. 2,1 B 0,2 1,0

Compare A B Looking Forward and Reasoning Backward A B h ~h r ~r Looking Forward and Reasoning Backward A B h ~h r ~r Reasoning Forward 2,1 2,1 0,2 0,2 1,0 1,0

Sequential Rationality COMMANDMENT Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today