The question isnt how much you bet against his bet but how much you are willing to pay him to play a given game with a fixed value for the outcomes (i.e winning losing and tieing). Note this is different from a bet in exactly the way you mention with the 3rd player, except that you pay your oppeont and not someone else - i.e you lost your payment even if you tie.
If you win 75% and him the rest and you get 1 for win 0 for tie and 1 for lose, you get 1 with 75% chance and lose 1 with 25% chance (0.5 on average) and should then be willing to give him upto 0.5 before playing the game because you then come up even (equvialently the game is fair/even if you get 0.5 for winning (1 minus the 0.5 you paid upfront), -0.5 for tieing and -1.5 for losing). Similarly if you pay him 0.5 and you win 1 with 50% chance and tie and get 0 with the remaning you are still even.
Game theory theories are math theories and not physics theories. Building on some given assumptions (i.e. what we measure is what you are willing to pay upfront) the theories are correct (if we look away from posible errors in the proofs - that said this is von Neumanns normal form games we are looking at right now and the proofs are correct under the assumptions used).