The odds should be 37.5% regardless. There's psychology involved, of course, but no more-so than Rock-Paper-Scissors. The fact that this is a multiplayer game also probably affects things, but I'm too lazy to figure that one out.
Let's assume there's only one player and one dealer. An arbitrary first round might look like this:
Both people picked a direction randomly so the Player had a 75% chance to survive.
Both players are now faced with a choice. The Player (P) can choose to
match the Dealer (D)'s first move, or
defect and pick a different option. Similarly, the Dealer can choose to
match the Player's first move or
defect and pick a different option. This gives four cases: Match-Match, Match-Defect, Defect-Match, and Defect-Defect.
(D)Match-(P)Match
Both players choose to match the other's previous move, so the 2nd round looks like this:
In the 3rd round, both players randomly pick from '<' or '>', giving the player a 50% chance of success. The 4th round is an automatic success if the 3rd is passed.
This gives a total success rate of 75% * 50% = 37.5%
(D)Match-(P)Defect
The dealer sacrifices the 2nd round be deliberately matching the Player's first move. The player chooses randomly from '<' or '>'. Let's say '<'.
In the 3rd round, the dealer can pick from (<, >) and the player can pick from (^, >). If the player picks '^', they lose if the dealer picks '<' causing '>' to be the only option in the 4th round. If the player picks '>' they lose if the dealer picks '>' as well. Either way, that's a 50% chance of success for a total success rate of 75% * 50% = 37.5%.
(D)Defect-(P)Match
The player stays safe in the second round and the dealer randomly picks from '<' or '>'. Let's say '<'.
In the 3rd round, the dealer can pick from (v, >) and the player can pick from (<, >). If the player picks '<', they lose if the dealer picks 'v' causing '>' to be the only option in the 4th round. If the player picks '>' they lose if the dealer picks '>' as well. Either way, that's a 50% chance of success for a total success rate of 75% * 50% = 37.5%.
(D)Defect-(P)Defect
Both players choose to defect, meaning both randomly choose from '<' or '>'. There's a 50% chance of them choosing differently, something like this:
The 3rd and 4th rounds are a formality; the player matches the dealer's two previous moves. Having to survive the 50% chance, however, leads to a total success of 75% * 50% = 37.5%.