The centipede game has two players who alternate in making decisions. At each turn, a player can choose between going "down" and ending the game or going "across" and continuing it (except at the last node where going "across" also ends the game). The longer the game goes on, the higher the total utility. However, a player who ends the game early will get a larger share of what utility there is. Here is one formulation of the problem:

Verify the game strategy where all choices are {0,1} besides the first two actions:

This can be understood by the fact the subgame perfect equilibrium is verified by only changing a single move. Thus, by maintaining the first two optimal actions, any such strategy is a subgame perfect equilibrium:

The tree game version of the Matching Pennies game is a game where each of two people chooses either Head or Tail. If the choices differ, person 1 pays person 2 a dollar; if they are the same, person 2 pays person 1 a dollar. Generate a tree game for the Matching Pennies game:

Find the formula of a subgame perfect equilibrium:

You may now find any number of subgame perfect equilibria:

Rock Paper Scissors is a zero-sum game, where either one player wins and the other loses, or there is a tie. Consider the tree version of this game, where the second player can choose an action considering the action of the first player:

Verify that the game strategy of the second player is always optimal whatever the strategy of the first player is. Since the conditions obtained below are true in the case of probability, it is the case that the game strategy of the second player is always optimal whatever the strategy of the first player is. Thus, the first player is at a disadvantage by playing first: