There is an increasing tendency for firms to use pricing algorithms that speedily react to market conditions, such as the ones used by major airlines and online retailers like Amazon.
I consider a dynamic model in which firms commit to pricing algorithms in the short run. Over time, their algorithms can be revealed to their competitors and firms can revise them, if they so wish.
I show how pricing algorithms not only facilitate collusion but inevitably lead to it.
To be precise, within certain parameter ranges, in any equilibrium of the dynamic game with algorithmic pricing, the joint profits of the firms are close to those of a monopolist.

Empirical analysis of discrete games relies on behavioral assumptions that are crucial not just for estimation, but also for the validity of counterfactual analyses and policy implications.
We find conditions to identify whether actual behavior satisfies these assumptions.
Our results allow one to identify whether and how often firms in an entry game play Nash equilibria, whether they enter simultaneously or sequentially, and whether their profit functions are private information or common knowledge.
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Example — identifying equilibrium conditions

Firms \(\ i=1,2\ \) decide whether to enter a market \(\ (y_i=1)\ \) or not \(\ (y_i=0)\).
Profits are as in Figure 1, where \(\ \beta_0 =(\beta_{0j})\ \) is a vector of unknown structural parameters, \(\ \mathbf{x} = (\mathbf{x}_i)\ \) is a vector of observable continuous firm characteristics with large support, and \(\ \mathbf{e} = (\mathbf{e}_i)\ \) is a vector of unobserved i.i.d. standard normal random errors independent of \(\ \mathbf{x}\).

Suppose the econometrician observes the joint distribution of \(\ \mathbf{x}\ \) and entry decisions.
Assuming that firms are profit maximizers and this fact is mutual knowledge, could she tell from the data whether firm's entry choices always constitute Nash equilibria?

The answer is “yes, she could!”

First, the large support of \(\ \mathbf{x}\ \) and the profit-maximization assumption identify \(\ \beta_0\ \) at infinity.
For example, as \(\ \mathbf{x}_1\to-\infty\), firm \(\ 1\ \) stays out of the market almost surely, and thus \[ \Pr\left(\mathbf{y}_2=1 \big|\mathbf{x}_1\right) = \Pr\left(\mathbf{e}_2 \leq \beta_{02}\mathbf{x}_2 \big| \mathbf{x}_1\right). \]
Because the left-hand-side probability is observed, this equation identifies \(\ \beta_{02}\).

Second, we have to assume that \(\ \mathbf{x}\ \) only affects behavior through payoffs.
Then, we can show that the facts that the covariates have continuous support and the error terms’ distribution belongs to the exponential-family imply that the disribution of choices conditional on both \(\ \mathbf{x}\ \) and \(\ \mathbf{e}\ \) is identified.

Therefore, the econometrician can recover both the profit functions and the firm's behavior as a function of both observed and unobserved heterogeneity.
This allows her to determine whether, how often and when firm's choices are in equilibrium.
In our paper, Nail and I generalize this approach to identify a wide class of behavioral and structural assumptions for a general class of discrete semi-parametric games of complete information.

We study the optimal design of a liability sharing arrangement between two agents as an infinitely repeated game.
Each period the active agent can take a costly, unobservable action to try to avert a crisis.
If a crisis occurs, both agents decide how much to contribute to mitigate it.
When the cost of the avoidace action is too high the first-best is not achievable.
In that case, at every contrained Pareto efficient PPE of the repeated game, the active agent “shirks” infinitely often, and when a crisis happens, the active agent is “bailed out” infinitely often.
The frequencies of crises and bailouts are endogenously determined at equilibrium.

Interdependent-choice equilibrium (ICE) is an extension of correlated equilibrium in which the mediator is able to choose the timing of her signals, and observe the actions taken by the players.
It characterizes all the outcomes that can be implemented in single shot interactions without repetition, side payments, binding contracts or any other form of delegation.
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Example — cooperaton in a prisoners’ dilemma

Two suspects of a crime are offered a sentence reduction in exchange for a confession.
While making his statement, each prisoners chooses whether to cooperate by remaining silent \(\ (\mathrm{C})\), or defect by signing a full confession \(\ (\mathrm{D})\).
Preferences are as in Figure 1.

\(\mathrm{C}\)

\(\mathrm{D}\)

\(\mathrm{C}\)

\(1,\ 1\)

\(-k,\ 1+g\)

\(\mathrm{D}\)

\(1+g,\ -k\)

\(0,\ 0\)

Fig 1 — Prisoners’ payoffs, whith \(\ g,k>0\)

The prisoners need not choose simultaneously nor independently.
There may be different ways to coordinate thier choices.
For instance, they could hire the same lawyer to schedule and be present in all the interactions with the DA and instruct her as follows:

Uniformly randomize the order of our meetings.
If the DA offers a deal, recommend that we reject it, unless one of us has already confessed, in which case recommend that we also confess.
Other than those recommendations, do not provide us with any additional information.

The resulting situation corresponds to the extensive form game in Figure 2.
As long as \(\ g\leq 1,\) the strategies represented with red arrows are a SPNE in which both prisoners cooperate.
This is possible because, along the equilibrium path, each prisoner is concerned that if he confesses then his accomplice will learn of his defection and will punish him by also confessing.

The prisoners can sustain cooperation in equilibrium, despite it being a single-shot interaction, and without using side payments, signing contracts or using any other form of binding commitment.

In the rest of the paper, I find a canonincal class of mechanisms that characterizes the set of all outcome distributions that can be implemented without commitment, not just for the prisoners’ dilemma, but for any finite strategic-form game.
The set of implementable outcomes admits a parsimonious representation in terms of a fnite set of affine inequalitites.

We extend Börgers (1993) by showing that pure and mixed dominance are equivalent when agents are sufficiently risk averse.
This implies that, in the absence of cardinal information, rationalizability is equivalent to the iterated removal of strategies that are dominated by a pure strategy.
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Example — a fair bet

A rational agent chooses between betting on an event \(\ E\), betting against \(\ E\), and not betting at all.
Her preferences are represented by the vNM indexes in Figure 1.

\(E\)

not \(E\)

bet on \(E\)

\(2\)

\(0\)

bet against \(E\)

\(0\)

\(2\)

do not bet

\(\gamma\)

\(\gamma\)

Fig 1 — Payoff matrix with \(\gamma\in(0,2)\)

Note that the ordinal ranking of state-action pairs remains unchanged as long as \(\ 0 < \gamma < 2\).
Also, not betting is not strictly dominated by any pure action, and, if \(\ \gamma \geq 1\), it is also not strictly dominated by any mixed action.
However, if \(\ \gamma <1\), then mixing uniformly between betting for and against \( E\) strictly dominates not betting.

Here, \(\ \gamma\ \) measures the degree of risk aversion of the agent.
Hence, dominance by pure actions coincides with dominance by mixed actions if the agent is sufficiently risk averse, and there exists a sufficiently risk-averse utility function which is compatible with the given ordinal preferences.

In the rest of the paper, Bulat and I show that these two observations continue to hold for a large class of decision problems under uncertainty with ordinal preferences.

We show that the set of equilibrium payoffs of repeated games with perfect monitoring and public randomization are polytopes, and we use this fact to efficiently implement the APS algorithm.
This paper was anticipated by the work of Abreu and Sannikov (2014).
Since this (preliminary and incomplete) version was written before this realization, I've decided to include it here anyway as a technical note.