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Game Theory

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

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

  3. 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.

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

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


  6. 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

  7. 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

  8. 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” h A ~h

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

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

  11. 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

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

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

  14. What would happen if I didn’t use rollback?

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

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

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

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

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