This paper provides necessary and sufficient conditions for the existence of a pure strategy Bertrand equilibrium in a model of price competition with fixed costs. It unveils an interesting and unexplored relationship between Bertrand competition and natural monopoly. That relationship points out that the non-subadditivity of the cost function at the output level corresponding to the oligopoly break-even price, denoted by D(pL(n)), is sufficient to guarantee that the market sustains a (not necessarily symmetric) Bertrand equilibrium in pure strategies with two or more firms supplying at least D(pL(n)). Conversely, the existence of a pure strategy equilibrium ensures that the cost function is not subadditive at every output greater than or equal to D(pL(n)).

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This paper presents a model of the penalty-kick game between a soccer goalkeeper and a kicker, in which there is uncertainty about the kicker's type (and there are two possible types of kicker). To find a solution for this game we use the concept of Bayesian equilibrium, and we find that, typically, one of the kicker's types will play a mixed strategy while the other type will choose a pure strategy (or, sometimes, a "restricted mixed strategy"). The model has a simpler version in which the players can only choose between two strategies (right and left), and a more complex version in which they can also choose a third strategy (the center of the goal). Comparing the incomplete-information Bayesian equilibria with the corresponding complete-information Nash equilibria, we find that in all cases the expected scoring probability increases (so that, on average, the goalkeeper is worse off under incomplete information). The three-strategy model is also useful to explain why it could be optimal for a goalkeeper never to choose the center of the goal (although at the same time there were some kickers who always chose to shoot to the center).JEL Classification: C72 (non-cooperative games), L83 (sports).

This paper uses the model proposed by Chiappori, Levitt and Groseclose (2002) to test for mixed-strategy Nash equilibria in a game between a kicker and a goalkeeper, in a situation of a soccer penalty kick. The main contribution of this paper is to develop an alternative testing methodology, based on the use of a simultaneous-equation regression approach that directly tests the equilibrium conditions. Using the same data than Chiappori, Levitt and Groseclose we find similar results than them, and we are also able to separately analyze the behavior of different types of players (classified according to the foot that they use to kick the ball, and to the strategies that they choose to mix).JEL Classification: C72 (non-cooperative games), L83 (sports), C35 (simultaneousequation discrete-regression models). ResumenEste artículo utiliza el modelo propuesto por Chiappori, Levitt y Groseclose (2002) para testear equilibrios de Nash en estrategias mixtas en un juego entre un pateador y un arquero, en una situación de tiro penal en el fútbol. La mayor contribución del trabajo es que desarrolla una metodología alternativa, basada en el uso de un enfoque de regresión con ecuaciones simultáneas que testea directamente las condiciones de equilibrio del juego. Usando los mismos datos que Chiappori, Levitt y Groseclose se hallan resultados similares, pudiendo también analizarse separadamente el comportamiento de distintos tipos de jugadores (clasificados según el pie que usen para patear la pelota, y las estrategias que elijan mezclar).Clasificación del JEL: C72 (juegos no cooperativos), L83 (deportes), C35 (modelos de regresión de ecuaciones simultáneas con variables discretas).Descriptores: tiros penales en fútbol, estrategias mixtas, equilibrio de Nash, regresión de ecuaciones simultáneas, test de Wald.

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