## Strategic choices in realistic settings

##### Abstract

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.