2. Philosophy - Stefano Franchi

G AME THEORETIC GAMES
The typical graphical representation of a game present in Game Theory and Economic
Behavior will make this point clear. Consider the children’s game of “Matching Pennies.”
In this game, the two players agree that one will be “even” and the other will be “odd.” Each
one then shows a penny. The pennies are shown simultaneously, and each player may show
either a head or a tail. If both show the same side, then “even” wins the penny from “odd;”
if they show different sides, “odd” wins the penny from “even. It would be natural to try to
analyze this game by proceeding as follows: “let’s imagine that ‘even’ plays Tails, then if
‘odd’ plays ‘tails’, ‘even’ wins, otherwise ‘odd’ wins.’ And similarly or the other case.
Let’s denote with +1 a win by ‘even’ and -1 a win by ‘odd’“. 18 This may bring to the fol-
lowing tree-like representation, in which each level stands for a move, in order to capture
the dynamic aspect:
Instead, the standard game-theoretic description is a matrix where each row represents
one player’s strategy (not move) and each column stands for the other’s, with each single
cell representing the final result:
Show Heads Show Tails + 1 Show Heads Show Tails
+ 1
Show Heads Show Tails
- 1
Even Strategy 1
Odd
“show heads”
Strategy 1
“show heads”
Strategy 2
“show tails”
+ 1 - 1
- 1 + 1
Strategy 2
“show tails”
18. This is actually von Neumann and Morgenstern first example of a matrix representation of a game in
normalized form, see John von Neumann and Oskar Morgenstern, The Theory of Games and Economic
Behavior (Princeton: Princeton UP, 1944) 94,111. Hereafter referred to as TGEB.
- 1
+ 1
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