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Forward induction and market entry with an endogenous outside option


We consider a two-player sequential game in which players first choose whether to engage in a productive (market game) or unproductive activity (contest game) and then, if both players have chosen to enter the market, they compete in prices. Both economic activities are linked because the rents in the contest game are a fraction of the market profits. Subgame perfection predicts competitive pricing and a battle-of-the-sexes reduced-form game with two asymmetric Nash equilibrium, where only one firm enters. Our experimental results reject the prediction based on backward induction but are easily explained by forward induction arguments. The payoffs from the rent-seeking activity (outside option) influence pricing behaviour and prices do not converge to marginal costs. When the size of the rent seeking activities is large, firms coordinate better on economic activities and, in the event of market competition, prices converge to full collusion.

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Data availability statement

The experimental data are available from the authors upon request.


  1. In a companion paper, Morales et al (2022), we analyse the balance between productive and unproductive activities in a three-firm experiment.

  2. In the mixed-strategy equilibrium, a duopoly in the market happens with probability \({\left(1-t\right)}^{2}\), a monopoly happens with probability \(2t\left(1-t\right)\) and no production occurs with probability \({t}^{2}\).

  3. We do not obtain sorting as predicted by some learning models (see Duffy and Hopkins 2005) nor do we obtain alternation (Erev et al. 2010) although there is some non-significant evidence of this phenomenon in the treatment High.

  4. We compare the entry rate to 50% because in a battle-of-the-sexes game, there are two asymmetric Nash equilibria. Assuming Nash predictions, miscoordination may originate from both players choosing different equilibria. If they randomise between both pure strategy equilibria, the level of miscoordination would be 50%. There is an additional symmetric mixed-strategy equilibrium in which players choose enter with probability \({\left(1-t\right)}^{2}\), and miscoordination occurs with probability \(1-2t\left(1-t\right)\). In both treatments, miscoordination would occur in equilibrium with probability 58%.

  5. The dummy variable High is statistically significant in none of the four models analysed. The reason is that, in the treatment High, only in 50% of the rounds we observe both firms entering the market and engaging in price competition, so we have far fewer observations of prices in the treatment High than in the treatment Low, where all firms enter the market to compete in prices. For example, in the second half of the experiment, there are two groups where firms never compete in prices (in the treatment High), so the number of groups for the second half of the experiment is 18.


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This study was funded by Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía (Grant nos. PY20-00069, P18-FR-3840).

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Correspondence to Antonio J. Morales.

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Appendix A: Translated instructions for treatment Low

The experiment will last for 30 rounds. Each round is independent (meaning that the results in a given round do not influence potential results in posterior rounds). At the beginning of the experiment you will be randomly matched with another participant; you will be part of a two-participant group. This matching will not change over the duration of the experiment. No participant will ever know the identity of the participant with whom he/she has been matched.

In each round you will have to make one or two decisions depending on your and your partner’s decisions. Your first decision is whether to enter or not a market. You will make this decision without knowing the decision of your partner.

  • If one of the members of your pair chooses to enter and the other does not, then the payoff for the member who entered is 70 whereas the payoff for the other person is 30.

  • In no one enters, then the payoff for each one is 0.

  • If both enter, then there is a second decision to be made: each one of you will have to choose a number between 0 and 100. Again, you will make this decision without knowing the decision of your partner.

    • If one of the numbers is lower than the other one, then the member of the pair who chose the lower number receives a payoff equal to that number, and the other member gets nothing.

    • If both numbers are equal, then each member of the pair receives half the chosen number.

At the end of each round you will be informed about your payoff and your partner’s choices and payoffs. A table with information of previous rounds will also be available.

At the end of the experiment the sum of your payoffs over the 30 rounds will be given to you privately using a conversion rate of 100 ECU = 1 €.

Appendix B: Further figures

See Figs. 5, 6, 7, 8 and 9.

Fig. 5
figure 5

Evolution of individual market entry rates by treatment (non-smoothed version)

Fig. 6
figure 6

Distribution of market outcomes over rounds by treatment

Fig. 7
figure 7

Evolution of average market prices over rounds by treatment

Fig. 8
figure 8

Evolution of the surplus captured by firms over rounds by treatment

Fig. 9
figure 9

Evolution of market and posted prices over rounds by treatment

Appendix C: Extensive form game

figure a

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Morales, A.J., Rodero-Cosano, J. Forward induction and market entry with an endogenous outside option. Soc Choice Welf 61, 365–383 (2023).

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