For a bet like this, everyone seeing it happen matters more than the odds.
Who buys the coffee? Who covers the next round? Rock-paper-scissors is quick for two people but drags on with six, and nobody wants to draw a ladder on a napkin. So people look for a game on the phone.
Put the names into Race Roulette and press Start; the characters run laps of a track. The result is drawn fairly before the start, and the race shows it — so lane position or luck on the track changes nothing.
It still keeps everyone watching to the end: runners finish 0.2–0.6 seconds apart, often close enough to need a photo finish; the lead changes more than ten times a race; and one race in three is decided in the last three seconds.
With equal odds, whoever paid last week can lose again. If that feels wrong, use weights. With a last-place-pays rule, give last time's payer more boosters: more boosters mean a better chance of finishing near the front, and so a smaller chance of coming last.
Boosters are drawn on screen, so nobody can claim someone got a quiet favour. The odds are exactly your weight ÷ the sum of all weights — the details are in how to run a draw people will accept.