Market and Games

Index

Introduction
Normal form games and Nash equilibrium
Nash equilibrium with mixed strategies
Applications to market competition
Extensive form games and backward induction
Subgame Perfect equilibrium (SPE)
Applications to vertical relations and mergers
Applications to innovation and R&D
Infinitely repeated games
Applications to collusion and cartels

Introduction

Game Theory: analyzing strategic interactions between agents where the outcome depends on the decisions of other agents.

The key assumptions is that agents act "rationally", meaning that:

agents choose actions that they believe is best for them while taking into account information available to them.
agents anticipate that other agents will also act rationally/strategically.

But what is rationality?

Rationality does NOT preclude the possibility of mistakes, errors, non-material values, altruistic or social preferences, social norms and cultures.

There are several ways to describe what an _economic model: is.

The Standard one: an economic model is a simplied (usually mathematical) description of reality, designed to yield hypotheses about economic behavior that can be tested.
The Fable one: economic theory formulates thoughts via "models". The word model sounds more scientific that the word fable or tale, but I think we are talking about the same thing. (Rubinstein, 2009)

In Game Theory, the approach lies between these 2 views:

we will develop and analyze simplified models that are certainly "fictional" (often with unrealistic assumptoins)
in most cases, predictions are consistent with intuitions under real-world policy and competition law.

Generally speaking, we can say that "all modelrs are wrong, but some are useful".

all models are abstractions, i.e. _intentionally neglet or simplify some aspects
a model is good if it has the right level of abstraction for a specific goal.

Indeed, unrealistic assumptionsallow a theory to focus only on the crucial and relevant elements required to obtaina prediction.

some research suggests that over-simplified assumptions (parsimonious economic models) deliver better predictions than black box algorithms _(random forest and kernel regressions) when moving across different contexts.

Oversimplified models are often superior, source: The transfer performance of economic models, 2022.

Normal form games and Nash equilibrium

Each game theory's game requires:

2 or more individuals (players, agents)
players interascting by making choices that jointly determine an outcome
(usually) every player can affect the outcome but no player has full control of the outocme

A solution is a set of reccomendations about how to play the game, such taht no player will have an incentive to not follow the recomendation.

Let's try to formalize it: Formal Form Games with pure strategies

An $n$ -player normal form game consists in:

for each player $i = 1, 2 \dots n$ , a set of strategies $S_{i}$
for each player $i = 1, 2 \dots n$ , a utility function $u_{i}$ that, for each strategy profile $(s_{1}, s_{2} \dots s_{n}) \in S_{1} \times \dots \times S_{n}$ specified a real number: $u_{i} (s_{1} \dots s_{n}) \in R$ .

We denote the game by $G = (S_{1}, \dots S_{n}, u_{1} \dots u_{n})$ .

A Nash Equilibrium is a strategy profile such that no player has an incentive to unilaterally deviate from their corresponding strategy.

Let $S$ be the set of strategy profiles, i.e. $S = S_{1} \times \dots \times S_{n}$ .

Given a $n$ -player normal form game $G$ , a strategy profile $s^{*} = (s_{1}^{*}, \dots s_{n}^{*})$ is a Nash Equilibrium of $G$ if for each player $i = 1, \dots n$ :

u_{i} (s_{1}^{*}, \dots s_{i - 1}^{*}, s_{i}^{*}, s_{i + 1}^{*}, \dots s_{n}^{*}) \geq u_{i} (s_{1}^{*}, \dots s_{i - 1}^{*}, s_{i}^{,} s_{i + 1}^{*}, \dots s_{n}^{*}) \forall s_{i} \in S_{i} π

The asterisk $^{*}$ is often used to denote an equilibrium profile rather than a generic profile.
When strategy $s_{i} \neq = s_{i}^{*}$ , it is called "unilateral deviation" from $s^{*}$ .

An alternative definition of Nash Equilibrium is: a strategy profile $(s_{1}, \dots s_{n}$ ) is NOT a Nash Equilibrium if at least one player has an incentive to unilaterally deviate.

We now consider a $n$ -player normal form game $G = {S_{1}, \dots S_{n}; u_{1}, \dots u_{n}}$ and a strategy profile $(s_{1}, \dots s_{n})$ .

Definitin: Best response in pure strategies

The set of player $i$ 's best responses against $(s_{1}, \dots, s_{i - 1}, s_{i + 1}, \dots s_{n})$ is the set of player's strategies that solve:

s_{i}^{'} \in S_{i} max u_{i} (s_{1}, \dots, s_{i - 1}, s_{i}^{'}, s_{i + 1}, \dots s_{n})

We denote as $R_{i} (s_{1}, \dots, s_{i - 1}, s_{i + 1}, \dots s_{n})$ the set of player $i$ 's best responses against $(s_{1}, \dots, s_{i - 1}, s_{i + 1}, \dots s_{n})$ .

Theorem: Nash Equilibrium in pure strategies

A strategy profile $(s_{1}^{*}, \dots s_{n}^{*})$ is a Nash Equilibrium if an omly if, $\forall i = 1, \dots n$ , we have:

s_{i}^{*} \in R_{i} (s_{1}^{*}, \dots, s_{i - 1}^{*}, s_{i + 1}^{*}, \dots s_{n}^{*})

In mathematics, we would call this a fixed point.

WHen we are analyzing the strategic interactions in both theoretical and real-orld settings, we are often interested in "efficiency".

Altought there are many different definition for it, the most standard one is called "Pareto-efficiency".

An outocome is Pareto-efficient if there is an outcome that makes an individual better off without making another individual worse off. In other words:

Definition: Pareto Efficiency

A strategy profile $s = (s_{1}, \dots s_{n}) \in S$ is Pareto inefficient if there is a strategy profile $s^{'} \in S$ such that:

all agents are weakly better off: $u_{i} (s^{'}) \geq u_{i} (s)$ for sall $i$ AND
at least one agent $j$ is strictly better off: $u_{j} (s) \geq u_{j} (s)$ .

An outcome that is not pareto inefficient is Pareto efficient.

Nash equilibrium with mixed strategies

So far, we assumed that players only choose "pure" strategies (non-random or deterministic). But, more generally, we could consider mixed (random) strategies , to ensure that a Nash Equilibrium exists for a broder class of games.

Consider for example the following example. Two players can choose between $H$ , $T$ and obtain the respect payoff.

It can be shown that there's no Nash Equilibrium with pure strategies.

Mixed Strategies

So let's suppose that player $1$ chooses $H$ with probability $x$ and $T$ with probability $1 - x$ . The same applies for player $2$ : he chosses $H$ with probability $y$ , $T$ with probability $1 - y$ .

We want to find $x, y \in [0, 1]$ such that it is a Nash Equilibrium.

Given that player $1$ chooses $H$ with probabilkty $x$ , Player's $2$ payoffs are:

$Player 2’s payoff from H = - x + (1 - x) = 1 - 2 x$
$Player 2’s payoff from T = x + (- 1) (1 - x) = 2 x - 1$

Player $2$ stricly prefers $H$ if:

1 - 2 x > 2 x - 1 ⟹ x < 1/2

We can write Player $2$ choices in a more compact way:

R_{2} (x) = ⎩ ⎨ ⎧ y = 1 0 \leq y \leq 1 y = 0 if x < 1/2, if x = 1/2, if x > 1/2,

Simirarly, given that Player $2$ chooses $H$ with probability $y$ , we can write:

$Player 1’s payoff from H = y + (- 1) (1 - y) = 2 y - 1$
$Player 1’s payoff from T = (- 1) y + (1) (1 - y) = 1 - 2 y$

R_{1} (y) = ⎩ ⎨ ⎧ x = 1 0 \leq x \leq 1 x = 0 if y > 1/2, if y = 1/2, if y < 1/2,

A Nash equilibrium occurs when the strategy profile is such every player is playing a best response to the other player’s strategy.

In other words, we have to find $x^{*}, y^{*}$ such that:

R_{1} (y^{*}) = x^{*}, R_{2} (x^{*}) = y^{*}

By plotting, we can easily identify them:

Mixed Strategies

The Nash Equilibrium is then $x^{*} = y^{*} = 1/2$ .

Mixed Strategies: formal definition

A mixed strategy for a player $i$ is a probability distribution $p_{i}$ over their set of pure strategies.

The set of all the pure strategies $S_{i}$ is denoted by $Δ (S_{i})$ .

In general, there' NO always a Nash Equilibrium with mixed strategies but, for a broad class of games, there's always a Nash Equilibrium.

This result was proven by John F. Nash, Jr. in his seminal 1950 paper “Equilibrium points in $n$ -person games”. Nash was awarded 1994 Nobel Prize in Economics for this and other contributions to game theory.

Consider then a $n$ -player normal form game $G = (S_{1} \dots S_{k}, u_{i} \dots u_{k})$ , with $n$ finite number and $S_{i}$ finite for every $i$ .

Theorem (Nash, 1950)

The game $G$ as at least one Nash equilibrium, possibly involving mixed strategies.

A couple of funny notes:

The paper is extremley short, just 3 pages!
There is a book and film about Nash’s life “A Beautiful Mind”

Important result #1: When checking for profitable unilaterally decisions, it is necessary and sufficient only pure strategy decisions $p_{i} = s_{i}$ , i.e. $p^{*}$ is a Nash Equilibrium only if:

U_{i} (p_{1}^{*}, \dots p_{i}^{*}, \dots p_{n}^{*}) > U_{i} (p_{1}^{*}, \dots s_{i}, \dots p_{n}^{*})

for every pure strategy $s_{i}$ .

Important result #2: Suppose $p = (p_{1}^{*}, \dots p_{n}^{*})$ is Nash Equilibrium. If players other than $i$ play according to $(p_{1}^{*}, \dots p_{n}^{*})$ , then player $i$ is indifferent between any two pure strategies that they play with strictly positive probability in $p_{i}^{*}$ .

Best response: formal definition

The set of player $i$ 's best responses against $(p_{1}, \dots p_{i - 1}, p_{i + 1}, \dots p_{n})$ is the set of strategies that solve:

s_{i} \in Δ (S_{i}) max U_{i} (p_{1}, \dots p_{i - 1}, p_{i}^{'}, p_{i + 1}, \dots p_{n})

We denote by $R_{i} (p_{1}, \dots p_{i - 1}, p_{i + 1}, \dots p_{n})$ the set of best responses against $(p_{1}, \dots p_{i - 1}, p_{i + 1}, \dots p_{n})$ .

Theorem (N.E. characterization via best responses)

A strategy profile $(p_{1}^{*}, \dots p_{n}^{*})$ is a Nash Equilibrium if and only if, $\forall i = 1, \dots n$ , we have:

p_{i}^{*} \in R_{i} (p_{1}^{*}, \dots p_{i - 1}^{*}, p_{i + 1}^{*}, \dots p_{n}^{*})

Intuition:

when looking for Nash Equilibrium:
- we calculate the player's best response functions
- find the intersection(s) of all players' best response functions
- translate the response functions into strategies.

Formal definition: strictly dominated strategies

Given a game $G$ , the mixed strategy $p_{i}$ is strictly dominated by $p_{i}$ if, for every pure strategy profile of the other players, we have:

U_{i} (s_{i}, \dots s_{i - 1}, p_{i}, s_{i + 1}, \dots s_{n}) < U_{i} (s_{i}, \dots s_{i - 1}, p_{i}^{'}, s_{i + 1}, \dots s_{n}) <

If a player has a stricly dominated strategy, he will never use it in a Nash Equilibrium.

Theorem: (Iterated deletion of strictly dominated strategies)

Let G be an $n$ -player normal form game such that player $i$ has a strictly dominated strategy in $G$ . Let $G'$ be the $n$ -player normal form game that is obtained from $G$ by removing such a strictly dominated pure strategy.

Then $G$ and $G'$ have the same set of Nash equilibria.

Formal definition: weakly dominated strategies

Instead of writing the full strategy profile of the other players as $(s_{i}, \dots s_{i - 1}, s_{i + 1}, \dots s_{n})$ , it is convenient to write as $s_{- i}$ .

Given a game $G$ , the mixed strategy $p_{i}$ is weakly dominated by $p_{i}^{'}$ if, for every pure strategy profile of the other players $s_{- i}$ , we have:

U_{i} (s_{- i}, p_{i}) \leq U_{i} (s_{- i}, p_{i}^{'})

and there exists at least one pure strategy profile $s_{i - 1}^{'}$ such that:

U_{i} (s_{- i}^{'}, p_{i}) < U_{i} (s_{- i}^{'}, p_{i}^{'})