Strategic choices in realistic settings
In this thesis, we study Bayesian games with two players and two actions (2 by 2 games) in realistic settings where private information is correlated or players have scarcity of attention. The contribution of this thesis is to shed further light on strategic interactions in realistic settings. Chapter 1 gives an introduction of the research and contributions of this thesis. In Chapter 2, we study how the correlation of private information affects rational agents’ choice in a symmetric game of strategic substitutes. The game we study is a static 2 by 2 entry game. Private information is assumed to be jointly normally distributed. The game can, for some parameter values, be solved by a cutoff strategy: that is enter if the private payoff shock is above some cutoff value and do not enter otherwise. Chapter 2 shows that there is a restriction on the value of correlation coefficient such that the game can be solved by the use of cutoff strategies. In this strategic-substitutes game, there are two possibilities. When the game can be solved by cutoff strategies, either, the game exhibits a unique (symmetric) equilibrium for any value of correlation coefficient; or, there is a threshold value for the correlation coefficient such that there is a unique (symmetric) equilibrium if the correlation coefficient is below the threshold, while if the correlation coefficient is above the threshold value, there are three equilibria: a symmetric equilibrium and two asymmetric equilibria. To understand how parameter changes affect players’ equilibrium behaviour, a comparative statics analysis on symmetric equilibrium is conducted. It is found that increasing monopoly profit or duopoly profit encourages players to enter the market, while increasing information correlation or jointly increasing the variances of players’ prior distribution will make players more likely to choose entry if the equilibrium cutoff strategies are below the unconditional mean, and less likely to choose entry if the current equilibrium cutoff strategies are above the unconditional mean. In Chapter 3, we study a 2 by 2 entry game of strategic complements in which players’ private information is correlated. As in Chapter 2, the game is symmetric and private information is modelled by a joint normal distribution. We use a cutoff strategy as defined in Chapter 2 to solve the game. Given other parameters, there exists a critical value of the correlation coefficient. For correlation coefficient below this critical value, cutoff strategies cannot be used to solve the game. We explore the number of equilibria and comparative static properties of the solution with respect to the correlation coefficient and the variance of the prior distribution. As the correlation coefficient changes from the lowest feasible (such that cutoff strategies are applicable) value to one, the sequence of the number of equilibrium will be 3 to 2 to 1, or 3 to 1. Alternatively, under some parameter specifications, the game exhibits a unique equilibrium for all feasible value of the correlation coefficient. The comparative statics of equilibrium strategies depends on the sign of the equilibrium cutoff strategies and the equilibrium’s stability. We provide a necessary and sufficient condition for the existence of a unique equilibrium. This necessary and sufficient condition nests the sufficient condition for uniqueness given by Morris and Shin (2005). Finally, if the correlation coefficient is negative for the strategic-complements games or positive for the strategic-substitutes games, there exists a critical value of variance such that for a variance below this threshold, the game cannot be solved in cutoff strategies. This implies that Harsanyi’s (1973) purification rationale, supposing the perturbed games are solved by cutoff strategies and the uncertainty of perturbed games vanishes as the variances of the perturbation-error distribution converge to zero, cannot be applied for a strategic-substitutes (strategic-complements) game with dependent perturbation errors that follow a joint normal distribution if the correlation coefficient is positive (negative). However, if the correlation coefficient is positive for the strategic-complements games or negative for the strategic-substitutes games, the purification rationale is still applicable even with dependent perturbation errors. There are Bayesian games that converge to the underlying complete information game as the perturbation errors degenerate to zero, and every pure strategy Bayesian Nash equilibrium of the perturbed games will converge to the corresponding Nash equilibrium of the complete information game in the limit. In Chapter 4, we study how scarcity of attention affects strategic choice behaviour in a 2 by 2 incomplete information strategic-substitutes entry game. Scarcity of attention is a common psychological characteristic (Kahneman 1973) and it is modelled by the rational inattention approach introduced by Sims (1998). In our game, players acquire information about their own private payoff shocks (which here follows a high-low binary distribution) at a cost. We find that, given the opponent’s strategy, as the unit cost of information acquisition increases a player’s best response will switch from acquiring information to simply comparing the ex-ante expected payoff of each action (using the player’s prior). By studying symmetric Bayesian games, we find that scarcity of attention can generate multiple equilibria in games that ordinarily have a unique equilibrium. These multiple equilibria are generated by the information cost. In any Bayesian game where there are multiple equilibria, there always exists one pair of asymmetric equilibria in which at least one player plays the game without acquiring information. The number of equilibria differs with the value of the unit information cost. There can be 1, 5 or 3 equilibria. Increasing the unit information cost could encourage or discourage a player from choosing entry. It depends on whether the prior probability of a high payoff shock is greater or less than some threshold value. We compare the rational inattention Bayesian game with a Bayesian quantal response equilibrium game where the observation errors are additive and follow a Type I extreme value distribution. A necessary and sufficient condition is established such that both the rational inattention Bayesian game and quantal response game have a common equilibrium.