Spoof Strategy
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This article presents an approach to the game of Spoof for 2 players, and is not meant to be mathematically rigorous. For an explanation of the game, see Spoof (game).
For any number of players, Spoof Strategy comprises two elements: mathematics and psychology. Both are more easily analysed for just two players – which is the most important element of the game since the 2-player game always occurs as the final and crucial round, when the loser is chosen.
In a 2-Player game, the minimum winning call is zero, when both players hold no coins, and the maximum is 6, when both players hold 3 coins. Note that in some games of Spoof, players are allowed to make ‘impossible’ (or ‘spoof’) calls: for example calling 6 when holding no coins. Although not analysed herein, this is not a winning strategy since basically the ‘spoof’ caller has no chance to win, and if not fooled his opponent still has at least 1 chance in 4 of winning when making a ‘possible’ call. Even if fooled, a player who believes he has no chance to win (as when he holds 3 coins and the first player has called 6) still must call, and can choose randomly between 3, 4, and 5 and will occasionally win if opponent is ‘spoofing’ in calling 6 with less than the anticipated 3 coins.
If faced with a first call of 6 when holding 3 coins the best second call is probably 3, putting the first caller on 0 – as far as possible from what was ‘shown’ when calling 6. Similarly, if holding 0 coins and facing a first call of 0 again the best second call is probably 3.
[edit] Mathematics of 2-Player Spoof
The first caller (FC) can make any one of 4 ‘possible’ calls, FC's own holding plus the estimate of the opponent’s holding, which may be 0, 1, 2 or 3. Thus if FC is holding 2 coins, the possible calls are 2, 3, 4 or 5... not 6, Brian. If no other information is available, each of the opponent’s possible holdings is equally likely, and so each of the possible calls is equally likely to win. The winning probability for the first caller is thus 1 chance in 4 or 0.25 or 25% no matter what FC holds or calls – so long as the call is possible.
This does not mean that all calls are equally likely to win in the end. Once the first player has called, the second caller (SC) has a call. Note that, as given above, FC has a 0.25 probability of winning, so SC may be already beaten, and the second call is pointless.
If not already beaten, SC may use any information gathered from FC’s call to decide the second call.
If FC calls 0 (or 6), FC can only be holding 0 (or 3) coins. If SC is not already beaten – by the 1 in 4 chance of SC also holding 0 (or 3) coins - SC can simply add 0 (or 3) to SC's own holding and make a winning call. SC winning chances are thus 3 in 4 or 0.75 or 75%. In this scenario, a draw is impossible. Outcomes: FC 25%, SC 75%, Draw 0%.
If FC calls 1 (or 5), FC can only be holding 0,1 (or 2,3) coins. If not already beaten, SC only has to choose between these two FC holdings – which gives a much better chance of winning than if SC had no knowledge of FC’s holding. If the guess is right, SC wins. If the guess is wrong, it is a draw or SC has already lost. The chances for SC are 1 in 2 or 0.5 or 50%. Outcomes: FC 25%, SC 50%, Draw 25%.
If FC calls 2 (or 4) FC can only be holding 0,1,2 (or 1,2,3) coins. Again, SC only has to consider these 3 possible holdings. Outcomes: FC 25%, SC 33%, Draw 42%.
If FC calls 3, FC may be holding any of 0,1,2,3 coins – SC has gained no information, and has to choose between 4 FC holdings. Outcomes: FC 25%, SC 25%, Draw 50%.
It is thus always best to call 3 as the first caller, since this gives the opponent the least information and the least chance of winning on the second call. By calling 3 as FC, you minimise the probability of the opponent winning to 25% which is the same probability that you win, so the game reduces to an equal chance of winning or losing (no different from a coin toss). If you are given the opportunitity to be the first caller or second caller, it is always optimal to choose to be the second caller as your opponent may not choose the most optimum call of 3 (by being FC and calling 3 you only have a 25% chance where as if you SC, you may have a better than 25% chance against a sub-optimal player who doesn't call 3). However, against an optimal random player who always calls 3, there is no advantage in choosing to be the first or second caller.
[edit] Elementary Psychology
If first caller will always call 3, how many coins should first caller hold? The answer is 3 minus the guess of opponent’s holding. If opponent always holds 2 coins, first caller should take out 1 coin and call 3. If opponent always holds 0 coins, take out 3 coins and call 3.
Of course, it would be a pretty dumb opponent who always took out the same number of coins. Optimum play is to be completely random in the number of coins chosen – but humans are not random beings and careful study can provide clues to opponent’s likely holdings and hence what to hold as first caller to make the desired total of 3.
Many players have a marked preference for ‘extreme’ holdings – 0 or 3 coins. Some will become ‘extreme’ (or stop being ‘extreme’) given the tension of being down to the last 2 players. Such tendencies are easy to exploit, as are compulsive ‘average’ holders, who will almost always take out 1 or 2 coins, especially when under stress.
Spotting such tendencies can also help in deciding what to hold as second caller. As an example, if opponent (now first caller) is aware of the mathematical strategy and always calls 3, whilst at the same time tending to hold a low number of coins (0 or 1, maybe 2, rarely 3) then it makes sense for second caller to hold 0 to maximize first caller’s likely error – but not every time.
When trying to analyse an opponent and adjust holdings accordingly, coins are no longer being chosen randomly and an alert player may be able to exploit this. If in doubt, select as randomly as possible.
Finally, some players allow their coins to “chink” leaving only a guess between their holding 2 or 3 coins. Others make a different fist when holding 0 coins. Such players do not last long, unfortunately.
