In game theory, Maximin/Minimax are strategies used when playing games. The strategy in this case is to maximize the smallest possible payoff that a player can receive as opposed to trying to maximize your payoff, assuming the other player will play rationally.

Consider the following game:

If this game was played with both players acting rationally, the optimal solution is for them both to invest. This is because "Invest" is a dominant strategy for P2. We can see this by finding P2's best response to P1.

- If P1 uses the strategy "Don't Invest" it is best for P2 to play "Invest" since $15 > $10.
- If P1 uses the strategy "Invest" it is best for P2 to play "Invest" since $20 > $0.

Therefore it always the best strategy for P2 to play "Invest". Now, given that P2 would know this (using the same logic that we both did), they just need to compare the options of "Invest" v "Don't Invest" assuming that P2 plays "Invest". Since $20 > $15, we can conclude the optimal solution is for both players to use the strategy "Invest" and thus they will both receive a payoff of $20.

Let's suppose now that P1 decides to use a Maximin strategy but P2 continues to play rationally. In this case, we play the game as follows:

- If P1 invests, the worst scenario is P2 doesn't invest in which case player 1 gets a payoff of -$100.
- If P1 doesn't invest, the worst case scenario is that P2 doesn't invest in which case P1 gets a payoff of $10.

Since $10 > -$100, playing the strategy "Don't Invest" maximizes the minimum payoff for P1, so P1 plays "Don't invest".

Now since firm B's dominant strategy is to play invest regardless of P1 strategy, they will play the strategy "Invest". The payoff in this situation will be $15 to P1 and P2. The new equilibrium is P1 playing "Don't invest" and P2 playing "invest". As we can see, from pursuing the strategy of Maximin, this has cost both players $5.

It is also worth noting that even if P2 used a Maximin strategy they would still choose to invest since that is a dominant strategy, which means it maximizes their payoff regardless of the other players actions. Clearly this also maximizes their minimum payoff.

This strategy is typically only used when one player believes that another player will not play rationally. The fear for P1 would be that if they play the strategy "Invest" and the other plays irrationally and chooses "Don't Invest", P1 would lose $100.

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