Suppose that 100 coins are flipped, and for every head \$1 enters a bucket. You and a friend are bidding on the bucket in a first price sealed bid auction, where bids are limited to integer values (and ties are solved uniformly). It is straightforward to see that the optimal strategy for both players is to bid \$49 (both players have an expected payoff of \$0.50.
Now suppose your opponent can see the result of the first 10 coin flips. Clearly they will now bid $\$(45+n)$, where $n$ is the number of heads they saw in the first 10 flips. What is your optimal strategy however? Should you still bid \$49? And how does the expected value change?