Game Theory¶

Auction¶

Consider \(n\) risk-neutral bidders, with independent private value \(v_i ~ F\)

Each bidder knows their own \(v_i\) and distribution \(F\), but not the \(v_i\) of others

Observed winning bids are the Bayesian Nash equilibrium outcome of each game $$ \begin{aligned} b_i &= v_i - \dfrac{1}{F(v_i)^{n-1}} \int \limits_0^{v_i} F(x)^{n-1} \cdot dx \ &= v_i - \dfrac{1}{n-1} \dfrac{G_n(b_i)}{g_n(b_i)} \end{aligned} $$ where

• \(b_i\) is the bid amount
• \(g_n\) is pdf of bid distribution
• \(G_n\) is cdf of bid distribution

There is no confounding between \(N\) (the number of bidders) and \(b_\max\) (the winning bid). Hence \(f (N) ≡ E [b_\max|N]\)non-parametrically identifies the effect of N on \(b_\max\)

The estimation problem is to learn \(f (N)\) from data. Here, theory helps specify the functional form of \(f (N)\) and therefore serves as a model selection mechanism

Theory also helps us to learn the values of the bidders – which cannot be identified nonparametrically – by specifying the functional form of the mapping from \(\{ v_i \}\) to \(\{ b_i \}\)

Furthermore, structural modelling will allows us to obtain other things as well such as: 2^nd-highest bid, etc

Structural Estimation¶

1. For each auction, non-parametrically estimate \(g_n\) and \(G_n\) from observed bids \(b_1, \dots, b_n\)
2. For each bidder, calculate

1. Estimate \(\hat F\) using \(\hat v_i\)
2. Predict winning bid

Monopoly¶

In each market \(m\) with population \(N_m\) and mean income \(I_m\), consumers choose between monopoly product and an outside good. Individual utilities are given by:

For each market \(m\), given demand \(q_m(p)\), the monopoly firm chooses \(p\) that maximizes its revenue $$ \begin{aligned} u_{i0}^m &= \epsilon_{i0}^m \ u_{i1}^m &= \beta_0 + \beta_1 I_m - \beta_2 p_m + \epsilon_{i1}^m \ \pi_m &= \sigma(\beta_0 + \beta_1 I_m - \beta_2 p_m) \ &=\dfrac{1}{1 + \dfrac{1}{\exp(\beta_0 + \beta_1 I_m - \beta_2 p_m)}} \ p &= \max_p { p \times q_m(p) - c(q_m(p)) } \ c' (q_m) &= p_m + [q'_m (p_m)]^{-1} q_m \end{aligned} $$ where

• \((u_{i0}^m, u_{i1}^m)\) are indirect utilities of outside good and monopoly product resp
• \(\epsilon_{ij}^m \sim \text{Gumbel}(0, 1)\)
• \(q_m \sim \text{Binomial}(N_m, \pi_m)\)
• \(c(q)\) is the monopoly firm’s cost function

Estimated marginal cost and demand curves for a market with median income and population

Here, theory helps us to learn the marginal cost function of the monopoly firm as well as the consumer utility function – neither of which is observed and neither can be nonparametrically identified.

Using the estimation results, we can conduct welfare analysis and make normative statements: For example, calculating the total deadweight loss due to monopoly

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