- Consider the following $2$ player finite extensive game. Player $1$ moves first. She must choose an integer from $1$ to $10$, and place that number of matchsticks on the table. Next, Player $2$ must add to this by picking an integer from $1$ to $10$, and adding that number of matchsticks to the ones placed by Player $2$. This game goes on until the player places matchsticks such that the total number reaches $30$, hence winning the game.
- Describe what choices Player $1$ must make to win this game, what number must she start with; then given the choice of Player $2$, what number must she pick and so forth.
For the above question, I found that playing $8$ in the first round would ensure that Player $1$ wins the game, given he plays rationally. I used the logic of backward induction, very vaguely though. In the last stage, for Player $1$ to win, he must ensure that the sum of matchsticks on the table is greater than or equal to $20$, thus ensuring that Player $1$ can choose a number between $1$ to $10$ such that sum becomes $30$ and he wins. For this, it must be the case that in the second last stage of the game, player $2$ chooses a number which can never be less than $19$, so that the only available options for him would be to make the sum greater than or equal to $20$, but not more than $29$. In the previous stage to the above Player $2$ move stage, Player $1$ must ensure that he makes the sum $19$. Thus moving backwards, I came to the conclusion that playing $8$ guarantees a win for Player $1$. If Player $1$ plays $8$ in the first round, Player $2$ can only make the sum greater than $8$ and less than $19$, such that in next stage, Player $1$ can choose such a number so as to make the sum $19$. Now Player $2$ can choose any number and lose since the sum will never be greater than $29$.
This is the only winning strategy I could find for Player $1$. I am not sure if this is correct or whether there are more winning strategies. Please help me in finding the complete winning strategy for Player $1$. Also, how do we formally write down the strategy (winning) set? $8$, $11-s_j^1$, $11-s_j^2$ is the only way I could think of writing the strategy set. Please correct me. Thanks a lot!