Two finger Morra

Two finger morra is one of the simplest examples of a strategic form game- there are two players, it is zero-sum, and the payoff matrix is 2×2, meaning each player chooses one of two options. The rules are related to an ancient Roman guessing/gambling game, micare digitis (to flash with fingers), which is described in the writeup on Morra; however, in this simple formulation, it is the parity of the total, rather than a successful guess of its value, which determines the victor.

To play a round, each player reveals either 1 or 2 fingers, with the winnings being the total number of fingers shown. If the total is odd, Player 1 wins; otherwise Player 2 wins. In strategic form for a 2 player zero sum game, the payoff matrix A is hence given by

-2   3
3   -4

The purpose of studying the strategic form is to attempt to determine a strategy for each player that is in some sense optimal. The ideal scenario for Player 1 is that there is a strategy that always enables him to win any given play of the game. However, such games are likely to be few and far between, and willing participants for the role of Player 2 even rarer.

However, we can more usefully tackle a refinement of this question, namely determining whether there is a strategy for Player 1 which, in the long run, they might expect to profit from. For any play of the game, Player 1 can pick either option 1, or option 2. Their return will then depend on the strategy employed by Player 2 for that particular play. We may suppose that Player 1 chooses option 1 with probability p₁, either in accordance with some plan or simply by chance; choosing option 2 the rest of the time, i.e., with probability p₂ = 1 - p₁. We will refer to this as a mixed strategy, although it is worth noting the special case of a pure strategy, where {p, q} = {0, 1} and thus only one option is ever used.

The objective for Player 1, therefore, is to devise a mixed strategy that maximises their payoff. At the same time, Player 2 is trying to minimise the payoff of Player 1, since this maximises their own payoff (by the zero-sum condition). Whilst neither player is aware of the particular option the other intends to take in a given play of the game, their calculations can take into account this motivation on the part of their opponent. Perhaps surprisingly, this does not descend into endless second-guessing, and should Player 1 find an optimal mixed strategy, they can even safely pre-declare the mix (although not a given move) without giving an advantage to Player 2. We shall illustrate how this arises for 2 finger Morra.

Note, however, that we have sidestepped some considerations of utility theory in our acceptance of expected payout as a good measure of the worth of a game, especially if the number of plays is to be small.

A strategy for Player 1

Can Player 1 guarantee a certain minimum (and preferably positive) payoff? Note that if she employs the mixed strategy (p₁, p₂), then her return depends on the strategy of Player 2:

• If Player 2 opts for '1', then the return for Player 1 is -2 (if she played 1) or 3 (if she played 2). Thus on average, she may expect a payoff of -2p₁ + 3p₂
• If Player 2 opts for '2', then Player 1's expectation is 3p₁ - 4p₂

If, therefore, we seek an expected payoff of at least V regardless of Player 2's strategy, then we require

• 2p₁ + 3p₂ ≥ V
• 3p₁ - 4p₂ ≥ V

As a first attempt, consider the case of equality:

V = -2p₁ + 3p₂ = 3p₁ - 4p₂
7p₂ = 5p₁
7(1 - p₁) = 5p₁
7 = 12p₁
7/12 = p₁

Thus we have a mixed strategy ( 7/12 , 5/12 ) where the expected payoff is -2( 7/12 ) + 3( 5/12 ) = 1/12 = 3( 7/12 ) - 4( 5/12 ). So Player 1 can guarantee an expected return of 1/12 per play (over a large number of plays).

A strategy for Player 2

Analogously to the duality theorem in linear programming, we may determine whether it is possible for Player 1 to ensure a greater expectation by seeing whether Player 2 is able to cap their losses at the 1/12 per play presented above.

In fact, Player 2 can minimise their losses in this way (and thus Player 1 must be content with the value of 1/12 ) by the same strategy. For Player 2 the expected payoffs are 2p₁ -3p₂ when Player 1 opts for '1’ and -3p₁ + 4p₂ when she opts for '2’; so with a mixed strategy of ( 7/12 , 5/12 ) Player 2 expects -1/12 in either case.

So, on average, Player 1 values the game as being good for at least 1/12 per play, whilst Player 2 can ensure it is no worse than -1/12 per play for them, i.e., it is at best worth 1/12 to Player 1. This turns out to be general behaviour for two player zero-sum games, a result which is captured by the minimax theorem.

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Part of A survey of game theory- see project homenode for details and links to the print version.