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Here is what I understood

Using Backward Induction, I inferred that buyer offers a price, say, $P$ and the seller will sell only if $P \geq c(I)$. Setting the lowest possible Price that will ensure that Seller sells the good would mean that buyer offers $P=c(I)$

The payoff to Seller would be $U_S=P-C(I)-I$ Putting $P=C(I)$ gives $U_S=C(I)-C(I)-I$ which would be negative. Therefore, seller would choose not to invest, that is, $I=0$

The subgame perfect strategy is Seller chooses to not invest, and Buyer offers $P=C(I)$

Is this the correct way to write the Subgame Strategy? Also, for the Social Optimum, I understand that we are supposed to find optimal $U_B+U_S$, i.e., $P-C(I)-I+v-P$, but I don't know how to proceed. Help would be appreciated!

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