# Aradillas-López and Tamer (2008)

These slides are based on the July 2007 version of the following article:

Aradillas-López, Andrés, Andres and Elie Tamer (2008).
The Identification Power of Equilibrium in Games.
*Journal of Business and Economic Statistics* 26, 261–283.

Presentation by Jason Blevins, Duke Applied Microeconometrics Reading Group, December 4, 2007.

## Introduction

Examines the identification power of Nash equilibrium assumptions.

Compare results by dropping Nash equilibrium and using only

*rationalizability*.Three examples are considered:

2x2 game of

*complete*information (e.g., Bresnahan and Reiss (1991)),2x2 game of

*incomplete*information,First price auction with independent private values.

Given a random sample, what can we learn about a parameter of interest using only level-$k$ rationalizability.

## Equilibrium Concepts

In simultaneous-move games, players attempt to predict what their rivals will play and act accordingly.

A

**Nash equilibrium**occurs when players’ expectations are are consistent with their opponents’ actions.A

**Rationalizable**strategy is a best response to*some*profile of one’s opponents’ strategies.Nash $\subseteq $ Rationalizable.

## Behavioral assumptions

Players use proper subjective probability distributions in analyzing uncertain events,

Players are expected utility maximizers,

The rules and structure of the game are common knowledge.

## Rationalizability

A strategy profile for player $i$ is

**dominated**if there exists another strategy that is better regardless of what other players do.Given a profile of strategies of player $i$’s opponents, ${s}^{-i}$, a strategy ${s}^{i}$ for player $i$ is a

**best response**if it is better than any other strategy given ${s}^{-i}$.Let ${S}^{i}(0)$ denote the set of player $i$’s strategies.

We say ${s}^{i}\in {S}^{i}(0)$ is a

**level–1 rational strategy**for player $i$ if $\exists {s}^{-i}\in {S}^{-i}(0)$ such that ${s}^{i}$ is a best response. Let ${S}^{i}(1)$ denote the set of such strategies.We say ${s}^{i}\in {S}^{i}(1)$ is a

**level–2 rational strategy**for player $i$ if $\exists {s}^{-i}\in {S}^{-i}(1)$ such that ${s}^{i}$ is a best response.And so on…

## I. 2x2 Game of Complete Information

Standard 2x2 normal-form game.

Given a sample $\{{y}_{i1},{y}_{i2}{\}}_{i=1}^{N}$ of market structures (entry decisions) in $N$ independent markets.

We want to learn about the joint distribution of $({t}_{1},{t}_{2})$ as well as the parameters ${\alpha}_{1}$ and ${\alpha}_{2}$.

Assume that ${\alpha}_{1},{\alpha}_{2}\le 0$.

## I. Level–1 rationality

If ${t}_{1}+{\alpha}_{1}\ge 0$, then ${a}_{1}=1$ is a dominant strategy for player 1.

If ${t}_{1}<0$, ${a}_{1}=0$, then is a dominant strategy.

If ${t}_{1}+{\alpha}_{1}\le 0\le {t}_{1}$, then both ${a}_{1}=0$ and ${a}_{1}=0$ are level–1 rational. They are best responses, respectively, when player 2 plays 1 or 0.

Similarly for player 2.

## I. Level–1 Rationality: Predictions

- In the middle region on the right side, note that player 2 does not rationally consider that player 1 would never play 0 in level–1 rationality. This illustrates the sequential nature of rationality.

## I. Level–2 rationality

Consider the region $({t}_{1},{t}_{2})\in [-{\alpha}_{1},\mathrm{\infty})\times [0,-{\alpha}_{2}]$.

$\mathcal{R}(1)=\{\{1\},\{0,1\}\}$

Player 2 believes player 1 will play ${a}_{1}=1$ with probability 1.

${a}_{2}=0$ is a best response.

We can eliminate ${a}_{2}=1$ at level 2.

$\mathcal{R}(k)=\{\{1\},\{0\}\}$ for $k\ge 2$.

## I. Inference

- Suppose we know the outcome probabilities $P(0,0),\dots ,P(1,1)$.
- Object of interest: $\theta =({\alpha}_{1},{\alpha}_{2},F(\cdot ,\cdot ))$.
- $F(\cdot ,\cdot )$ is the joint distribution of $({t}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2})$.
- Level–1 rationality implies the following restrictions on $\theta $:

$$\begin{array}{rl}\mathrm{Pr}({t}_{1}\ge -{\alpha}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\ge -{\alpha}_{2})& \le P(1,1)\le \mathrm{Pr}({t}_{1}\ge 0,\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\ge 0)\\ \mathrm{Pr}({t}_{1}\le 0,\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\le 0)& \le P(0,0)\le \mathrm{Pr}({t}_{1}\le -{\alpha}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\le {\alpha}_{2})\\ \mathrm{Pr}({t}_{1}\ge -{\alpha}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\le 0)& \le P(1,0)\le \mathrm{Pr}({t}_{1}\ge 0,\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\le -{\alpha}_{2})\\ \mathrm{Pr}({t}_{1}\le 0,\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\ge -{\alpha}_{2})& \le P(0,1)\le \mathrm{Pr}({t}_{1}\le -{\alpha}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2}\ge 0)\\ \end{array}$$

- The identified set ${\Theta}_{I}$ is the set of all $\theta $ which satisfy these inequalities.
- The model
*point identifies*$\theta $ if ${\Theta}_{I}$ is a singleton.

## II. 2x2 Game of Incomplete Information

Now assume that ${t}_{1}$ and ${t}_{2}$ are private information.

Common prior assumption on the joint distribution of $({t}_{1},\phantom{\rule{thinmathspace}{0ex}}{t}_{2})$.

Players have

*beliefs*about their opponents’ actions, conditional on their own type: ${\mathbb{P}}_{{t}_{1}}\equiv \mathrm{Pr}({a}_{2}=1|{t}_{1}).$As before, assume that ${\alpha}_{p}\le 0$ for $p=1,2$.

The expected payoffs are now: $$U({a}_{i},{P}_{{t}_{i}})=\{\begin{array}{ll}{t}_{i}+{\alpha}_{i}{\mathbb{P}}_{{t}_{i}}& \text{if}\phantom{\rule{1em}{0ex}}{a}_{i}=1\\ 0& \text{otherwise}\\ \end{array}$$

## II. Rationality

Consider only threshold strategies: ${Y}_{p}=1\{{t}_{p}\ge {\mu}_{p}\}$ for $p=1,2$.

Beliefs are thus probability distributions for ${\mu}_{-p}$ given ${\mathcal{I}}_{p}$ (which includes ${t}_{p}$): ${\hat{G}}_{p}({\mu}_{-p}|{\mathcal{I}}_{p}).$ This assumption effectively reduces the space of possible strategies to $\mathbb{R}$.

The concept of

*rationality*now has to account for*level-$k$ rational beliefs*. Beliefs are not required to be “correct.” Compare this with BNE beliefs, where all players know them to be correct.Level-$k$ rationalizable beliefs ${\hat{G}}_{p}({\mu}_{-p}|{\mathcal{I}}_{p})$ assign zero probability to strictly dominated strategies by player $-p$.

A strategy by player $p$ is level-$k$

*rationalizable*if it is a best response given level-$k$ rationalizable beliefs:

$${Y}_{p}=1\{{t}_{p}+{\alpha}_{p}{\int}_{\mathbb{S}({\hat{G}}_{p})}E[1\{{t}_{-p}\ge \mu \}|{\mathcal{I}}_{p},\mu ]\phantom{\rule{thinmathspace}{0ex}}d{\hat{G}}_{p}(\mu |{\mathcal{I}}_{p})\ge 0\}$$

- The support $\mathbb{S}({\hat{G}}_{p})$ of by applying iterated elimination of dominated strategies.

## II. Level–1 rationality

For

*any*belief function, the following must hold eventwise: $$1\{{t}_{p}+{\alpha}_{p}\ge 0\}\le 1\{{Y}_{p}=1\}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}1\{{t}_{p}<0\}\le 1\{{Y}_{p}=0\}$$All other decision rules are strictly dominated for all possible beliefs.

The above inequalities imply $$\mathrm{Pr}({t}_{p}+{\alpha}_{p}\ge 0)\le \mathrm{Pr}({t}_{p}\ge {\mu}_{p})\le \mathrm{Pr}({t}_{p}\ge 0)$$ or simply, ${\mu}_{p}\in [0,-{\alpha}_{p}]$.

This is the set of

*level–1 rationalizable strategies.*

## II. Level–2 rationality

Level 2 rationalizable beliefs:

assign zero probability to strictly dominated strategies ${\mu}_{p}\notin [0,-{\alpha}_{p}]$,

satisfy $\hat{G}(0|{\mathcal{I}}_{p})=0$ and $\hat{G}(-{\alpha}_{p}|{\mathcal{I}}_{p})=1$.

Level 2 rationalizable strategies are:

level 1 rationalizable (i.e., $0\le {\mu}_{p}\le -{\alpha}_{p}$),

best responses given level–2 rational beliefs.

A strategy ${Y}_{p}=1\{{t}_{p}\ge {\mu}_{p}\}$ is

*level–2 rationalizable*if $${\mu}_{p}=-{\alpha}_{p}{\int}_{0}^{-{\alpha}_{-p}}E[1\{{t}_{-p}\ge \mu \}|{\mathcal{I}}_{p},\mu ]\phantom{\rule{thinmathspace}{0ex}}d{\hat{G}}_{p}(\mu |{\mathcal{I}}_{p})$$

## II. Level-k rationality

We can summarize the set of level-$k$ rationalizable strategies in the class of threshold strategies ${Y}_{p}=1\{{t}_{p}\ge {\mu}_{p}\}$ as follows:

For $k=1$ and $p\in \{1,2\}$, $${\mu}_{p}\in [{\mu}_{p,k}^{L},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p,k}^{U}]\equiv [0,-{\alpha}_{p}]$$

For $k>1$ and $p\in \{1,2\}$, $${\mu}_{p}\in [{\mu}_{p,k}^{L},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p,k}^{U}]\equiv [-{\alpha}_{p}E[1\{{t}_{-p}\ge {\mu}_{-p,k-1}^{U}\}|{\mathcal{I}}_{p}],\phantom{\rule{thinmathspace}{0ex}}-{\alpha}_{p}E[1\{{t}_{-p}\ge {\mu}_{-p,k-1}^{L}\}|{\mathcal{I}}_{p}]]$$

Note that $[{\mu}_{p,k}^{L},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p,k}^{U}]\subseteq [{\mu}_{p,k-1}^{L},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p,k-1}^{U}]$ for any $k>1$

*a.s.*Any level-$k$ rational player is also level-$k\prime $ rational for any $1\le k\prime \le k-1$.

If there is a unique BNE $({\mu}_{1}^{\star},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{2}^{\star})$, then $$\underset{k\to \mathrm{\infty}}{\mathrm{lim}}[{\mu}_{p,k}^{L},\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p,k}^{U}]=\{{\mu}_{p}^{\star}\}.$$

## II. A Parametric Model

Let ${t}_{p}={X}_{p}^{\top}{\beta}_{p}-{\epsilon}_{p}$ for $p\in \{1,2\}$ where ${X}_{p}$ is observable to the researcher but ${\epsilon}_{p}$ is not. We wish to estimate ${\beta}_{p}$.

For simplicity, ${\epsilon}_{1}$ and ${\epsilon}_{2}$ are independent and ${\epsilon}_{p}\sim {H}_{p}(\cdot )$.

Player $p$ knows ${\epsilon}_{p}$ and believes ${\epsilon}_{-p}\sim {H}_{-p}(\cdot )$.

We now proceeds as follows:

Develop an objective function which can be used to construct the identified set ${\Theta}_{I}$,

Discuss identification of $k$,

Provide sufficient conditions for point identification.

## II. Iterative Construction of Beliefs

We no longer restrict ourselves to threshold strategies.

Let $\mathcal{I}$ be the information set of both players as well as the econometrician (

*i.e.*, the regressors).We iteratively construct bounds on the beliefs $\mathrm{Pr}({Y}_{-p}|\mathcal{I})$: $${\pi}_{-p}^{L}(\theta |k,\mathcal{I})\le \mathrm{Pr}({Y}_{-p}|\mathcal{I})\le {\pi}_{-p}^{U}(\theta |k,\mathcal{I}).$$

Initialize ${\pi}_{-p}^{L}(\theta |k=1,\mathcal{I})=0$ and ${\pi}_{-p}^{U}(\theta |k=1,\mathcal{I})=1$.

Then for each $k>1$ and each $p\in \{1,2\}$: $$\begin{array}{c}{\pi}_{p}^{L}(\theta |k,\mathcal{I})={H}_{p}({X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{U}(\theta |k-1,\mathcal{I})),\\ {\pi}_{p}^{U}(\theta |k,\mathcal{I})={H}_{p}({X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{L}(\theta |k-1,\mathcal{I})).\end{array}$$

It follows that $$[{\pi}_{-p}^{L}(\theta |k,\mathcal{I}),{\pi}_{-p}^{U}(\theta |k,\mathcal{I})]\subseteq [{\pi}_{-p}^{L}(\theta |k-1,\mathcal{I}),{\pi}_{-p}^{U}(\theta |k-1,\mathcal{I})]\phantom{\rule{1em}{0ex}}\text{a.s.}$$

## II. Iterative Belief Construction Example

## II. Finding the Identified Set

Player $p$ is $k$-rational if and only if $$1\{{X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{U}(\theta |k,\mathcal{I})\ge {\epsilon}_{p}\}\le 1\{{Y}_{p}=1\}\le 1\{{X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{L}(\theta |k,\mathcal{I})\ge {\epsilon}_{p}\}$$

We can use this relationship as a basis for inference.

Given some $k$, let ${W}_{p}\equiv ({X}_{p},\mathcal{I})$ and let $a,b\in {\mathbb{R}}^{dim({W}_{p})}$: $${\Lambda}_{p}(\theta |a,b,k)=E\left[(1-1\left\{\begin{array}{c}{H}_{p}({X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{U}(\theta |k,\mathcal{I}))\\ \le \mathrm{Pr}({Y}_{p}=1|{W}_{p})\\ \le {H}_{p}({X}_{p}^{\top}{\beta}_{p}+{\alpha}_{p}{\pi}_{-p}^{L}(\theta |k,\mathcal{I}))\end{array}\right\})1\{a\le {W}_{p}\le b\}\right].$$

Then define $${\Gamma}_{p}(\theta |k)=\iint {\Lambda}_{p}(\theta |a,b,k)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dF}}_{{W}_{p}}(a)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dF}}_{{W}_{p}}(b)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\Gamma (\theta |k)={({\Gamma}_{1}(\theta |k),\phantom{\rule{thinmathspace}{0ex}}{\Gamma}_{2}(\theta |k))}^{\top}$$

## II. Finding the Identified Set

We can use the previous expression to construct an objective function.

For some positive definite matrix $\Omega $, $$\Theta (k)=\{\theta \in \mathrm{arg}\underset{\theta}{\mathrm{min}}Q(\theta |k)\equiv \mathrm{arg}\underset{\theta}{\mathrm{min}}\Gamma (\theta |k{)}^{\top}\Omega \Gamma (\theta |k)\}.$$

By construction, $\Theta (k+1)\subseteq \Theta (k)$ for all $k$.

Given a random sample, we can use set inference methods to construct an estimator for $\Theta (k)$.

As compared with Nash equilibrium, we do not have to solve a fixed point problem. The iterative restrictions imposed by rationality are computationally simple by comparison.

For any $k$, $\Theta (k)$ is guaranteed to contain the BNE.

## II. Inference on the Rationality Level

A sample can also inform us about the level of rationality $k$.

Suppose all players are at most ${k}_{0}$-rational.

The level-${k}_{0}$ bounds should hold

*a.e.*but the level-$({k}_{0}+1)$ bounds may be violated.We can proceed as follows:

Construct $\Theta (1)$,

Define $\tilde{Q}(k)={\mathrm{min}}_{\theta \in \Theta (1)}Q(\theta |k)$,

$\tilde{Q}(k)=0$ for $k\le {k}_{0}$ but $\tilde{Q}(k)>0$ if $k>{k}_{0}$.

Thus, if $\tilde{Q}(k)>0$ and $\tilde{Q}(k-1)=0$ we can reject ${k}_{0}=k$.

## II. Point Identification Under Level–1 Rationality

Suppose players are level–1 rational.

Suppose ${X}_{p}$ has full rank for $p\in \{1,2\}$. Let ${X}_{d,p}$ and ${X}_{l,p}$ denote, respectively, regressors with bounded and unbounded support and let ${\beta}_{d,p}$ and ${\beta}_{l,p}$ be the corresponding coefficients. Finally, define $X\equiv ({X}_{1},{X}_{2})$.

${\beta}_{l,p}$ is identified if for each $p$, there is a continuous ${X}_{l,p}$ with nonzero ${\beta}_{l,p}$ and unbounded support conditional on ${X}_{-l,p}$ such that for any $c\in (0,1)$, $b\ne 0$ and $q\in {\mathbb{R}}^{dim({X}_{-l,p})}$, there exists ${C}_{b,q,m}>0$ such that $$\mathrm{Pr}({\epsilon}_{p}\le b{X}_{l,p}+{q}^{\top}{X}_{-l,p}|X)>m$$ for all ${X}_{l,p}$ and $sgn(b)\cdot {X}_{l,p}>{C}_{b,q,m}$.

## II. Point Identification Under Level–1 Rationality

${\beta}_{d,p}$ is identified if for any ${\alpha}_{p},{\beta}_{d,p},{\tilde{\beta}}_{d,p}\in \Theta $ with ${\tilde{\beta}}_{d,p}\ne {\beta}_{d,p}$, $$\mathrm{Pr}(|{X}_{d,p}^{\top}({\beta}_{d,p}-{\tilde{\beta}}_{d,p})|>\left|{\alpha}_{p}\right||{X}_{-d,p})>0.$$

If the above two properties hold, and for any $\Delta >0$, there exists ${\mathcal{X}}_{\Delta}\in \mathbb{S}({X}_{p})$ such that $\mathrm{Pr}({Y}_{p}=1|X)<\mathrm{Pr}({\epsilon}_{p}\le {X}_{p}^{\top}{\beta}_{{b}_{0}}+{\alpha}_{{p}_{0}}|X)+\Delta $ whenever ${X}_{p}\in {\mathcal{X}}_{\Delta}$, then the identified set for ${\alpha}_{p}$ is $\{{\alpha}_{p}\le {\alpha}_{{p}_{0}}\}$.

## III. First Price IPV Auction

Many symmetric, risk-neutral potential buyers.

Bids are made simultaneously for a single good.

Focus on independent private values (IPV) ${v}_{i}$.

Object of interest: ${F}_{0}(\cdot )$, the distribution of private values.

${F}_{0}(\cdot )$ is common knowledge among all players; has support $[0,\omega )$.

For simplicity, assume the reserve price is ${p}_{0}=0$.

## III. Interim Rationality

Point identification in BNE case established by Guerre, Perrigne and Vuong (2000).

Here, the equilibrium assumption is relaxed following Battigalli and Siniscalchi (2003).

Buyers under

*interim rationality*:rational,

expected utility maximizers,

strategically sophisticated to a particular degree $k$,

have beliefs that may or may not be “correct.”

## III. Assumptions on Beliefs

Bidders expect that

*any*positive bid will win with positive probability:No player will bid higher than ${v}_{i}$,

Each bidder with ${v}_{i}>0$ will submit a strictly positive bid.

The number of potential bidders $\mathcal{N}$ equals the number of actual bidders

*a.s.*

Beliefs only assign positive probability to

*increasing*bidding functions.The space of all such functions is $$\mathcal{B}=\{b(\cdot ):[0,\omega )\to {\mathbb{R}}_{+}|b(v)\le v\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}v>w\Rightarrow b(v)>b(w)\}.$$

## III. Level–1 Rationality

Player $i$’s problem is $$\underset{b\ge 0}{\mathrm{max}}({v}_{i}-b){\hat{\mathrm{Pr}}}_{i}[\underset{j\ne i}{\mathrm{max}}b({v}_{j})\le b].$$

Here, ${\hat{\mathrm{Pr}}}_{i}$ denotes player $i$’s beliefs about ${v}_{-i}$.

Level–1 rational bid satisfy $$b\le {v}_{i}\equiv {\overline{B}}_{1}({v}_{i},\mathcal{N}).$$

Any bid show

*any*bid below ${v}_{i}$ is level–1 rational (and thus we cannot bound bids from below).

## III. Level–2 Rationality

The worst case scenario for player $i$ is that for all $j$, $$b({v}_{j})={\overline{B}}_{1}({v}_{j},\mathcal{N})={v}_{j}.$$

Expected utility in this case is $$\underset{b\ge 0}{\mathrm{max}}({v}_{i}-b){\hat{\mathrm{Pr}}}_{i}[\underset{j\ne i}{\mathrm{max}}{\overline{B}}_{1}({v}_{j},\mathcal{N})\le b]=\underset{b\ge 0}{\mathrm{max}}({v}_{i}-b){F}_{0}(b{)}^{\mathcal{N}-1}\equiv {\pi}_{2}({v}_{i},\mathcal{N}).$$

${\pi}_{2}({v}_{i},\mathcal{N})$ is thus a lower bound over all beliefs. The upper bound is $({v}_{i}-b)$.

Thus, rational bids must satisfy $${v}_{i}-b\ge {\pi}_{2}({v}_{i},\mathcal{N})$$ or $$b\le {v}_{i}-{\pi}_{2}({v}_{i},\mathcal{N})\equiv {\overline{B}}_{2}({v}_{i},\mathcal{N}).$$

## III. Level-k Rationality

Proceeding recursively, one can show that at level-$k$, $${b}_{i}\le {v}_{i}-{\pi}_{k}({v}_{i},\mathcal{N})\equiv {\overline{B}}_{k}({v}_{i},\mathcal{N})$$ where $${\pi}_{k}({v}_{i},\mathcal{N})=\underset{b\ge 0}{\mathrm{max}}({v}_{i}-b){F}_{0}{({\overline{S}}_{k-1}(b,\mathcal{N}))}^{\mathcal{N}-1}$$ and $${\overline{S}}_{k-1}(b,\mathcal{N})\equiv {\overline{B}}_{k}^{-1}(\cdot ,\mathcal{N}).$$

We know that ${\overline{B}}_{k}(\cdot ,\mathcal{N})\ge {b}^{\mathrm{BNE}}(\cdot ,\mathcal{N})$ for all $k$. Furthermore, bidding below ${b}^{\mathrm{BNE}}$ is rationalizable for all $k$.

That is, the predictive power of $k$-rationality in this model is significantly less than that of BNE.

## III. Identification with Level–1 Rationality

Suppose ${F}_{0}$ lies in the space of log-concave, absolutely continuous distribution functions on $[0,\omega )$.

We are given a sample of $L$ auctions and want to recover $F$.

The model predicts that level–1 rational bids satisfy $0\le {b}_{i}\le {v}_{i}$ for all $i=1,\dots ,\mathcal{N}$.

The ${b}_{i}$’s are observed but the ${v}_{i}$’s are not, but we can still bound $F$: $${F}_{0}(t,\theta )\equiv \mathrm{Pr}(v\le t)\le \mathrm{Pr}(b\le t)\equiv {G}_{b}(t).$$

Empirical strategy: bound the distribution of valuations using the empirical distribution of bids.

## III. Identification with Level-k Rationality

Continue for $k>1$ using the bounds derived above.

We can derive an objective function as before: $$\begin{array}{c}\Lambda (\theta |a,c,k)=\int (1-1\{{F}_{b}(b)\ge F({\overline{S}}_{k}(b,\mathcal{N}|\theta ),\theta )\})1\{a\le b\le c\}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dF}}_{b}(b)\\ \Lambda (\theta |k)=\iint \Lambda (\theta |a,c,k)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dF}}_{b}(a)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dF}}_{b}(c)\\ \Theta (k)=\{\theta \in \Theta :\theta \in \mathrm{arg}\underset{\theta \in \Theta}{\mathrm{min}}\Gamma (\theta |k{)}^{2}\}\\ \end{array}$$

## Conclusion

Look at identification power of Nash equilibrium by comparing to level-$k$ rationality.

Derivation of identified set and corresponding objective functions for conducting inference.

Three cases:

2x2 game of complete information,

2x2 game of incomplete information,

first price auction.