This is yet another hat puzzle, but not one of the usual ones! Key differences/commonalities between other hat puzzles are emphasized.

Five friends are going to go on a game show. When the game begins, a red or blue hat will placed on each player's head, the colors randomly chosen by independent coin flips. They stand in a circle, and the stage is set so **each player can only see the hats of the players on their immediate right and left**. Everyone then **simultaneously** guesses red or blue. If **everyone** is correct, they win the jackpot; if anyone is wrong, they get nothing. 

As usual, before the game begins, they may devise a strategy, but after it begins, they cannot communicate in any way. 

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> Help these players find the best strategy, and convince them they can't do better.
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> 
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As a warmup, it may be helpful to try this puzzle with three players, instead of five. In that case, there is a strategy with probability $1/2$ of success (which is better than the $1/8$ chance they get by guessing randomly).