An empirical analysis of individual level casino gambling behavior

Abstract

Gambling and gaming is a very large industry in the United States with about one-third of all adults participating in it on a regular basis. Using novel and unique behavioral data from a panel of casino gamblers, this paper investigates three aspects of consumer behavior in this domain. The first is that consumers are addicted to gambling, the second that they act on “irrational” beliefs, and the third that they are influenced by marketing activity that attempts to influence their gambling behavior. We use the interrelated consumer decisions to play (gamble) and the amount bet in a casino setting to focus on addiction using the standard economic definition of addiction. We test for two irrational behaviors, the “gambler’s fallacy” and the “hot hand myth”—our research represents the first test for these behaviors using disaggregate data in a real (as opposed to a laboratory) setting. Finally, we look at the effect of marketing instruments on the both the decision to play and the amount bet. Using hierarchical Bayesian methods to pin down individual-level parameters, we find that about 8% of the consumers in our sample can be classified as addicted. We find support in our data for the gambler’s fallacy, but not for the hot hand myth. We find that marketing instruments positively affect gambling behavior, and that consumers who are more addicted are also affected by marketing to a greater extent. Specifically, the long-run marketing response is about twice as high for the more addicted consumers.

Appendices

Appendix A: Likelihood function

For ease of exposition, we use the following abbreviations in this appendix

$$ b_{it}=\ln\left(Bet_{it}\right) $$

(20)

$$ P_{it}=Play_{it} $$

(21)

$$ P_{it}^{*}=Play_{it}^{*} $$

(22)

Also, define the following

$$ d_{it}=\left(\begin{array}{c} b_{it}\\ P_{it}^{*} \end{array}\right) $$

(23)

$$ g_{it}=\left(\begin{array}{c} X_{it}\alpha_{i}\\ Y_{it}\beta_{i} \end{array}\right) $$

(24)

$$ \Sigma=\left(\begin{array}{cc} \sigma^{2} & \rho\sigma\\ \rho\sigma & 1 \end{array}\right) $$

(25)

$$ m_{\alpha}=\mbox{vec}\left(M_{\alpha}\right) $$

(26)

$$ m_{\beta}=\mbox{vec}\left(M_{\beta}\right) $$

(27)

The likelihood function can be written as follows

$${\cal L}\propto\begin{array}{c} \prod_{i=1}^{N}\left[\begin{array}{c} \prod_{t=1}^{T_{i}}\left[\begin{array}{c} \left[|\Sigma|^{-0.5}\exp\left(-\frac{1}{2}\left(d_{it}-g_{it}\right)'\Sigma^{-1}\left(d_{it}-g_{it}\right)\right)\right]^{P_{it}}\\ \left[\exp\left(-\frac{1}{2}\left(P_{it}^{*}-Y_{it}\beta_{i}\right)^{2}\right)\right]^{1-P_{it}}\\ \left[1\left(P_{it}^{*}>0\right)\right]^{P_{it}}\left[1\left(P_{it}^{*}<0\right)\right]^{1-P_{it}} \end{array}\right]\\ |V_{\alpha}|^{-0.5}\exp\left(-\frac{1}{2}\left(\alpha_{i}-M_{\alpha}z_{i}\right)'V_{\alpha}^{-1}\left(\alpha_{i}-M_{\alpha}z_{i}\right)\right)\\ |V_{\beta}|^{-0.5}\exp\left(-\frac{1}{2}\left(\beta_{i}-M_{\beta}z_{i}\right)'V_{\beta}^{-1}\left(\beta_{i}-M_{\beta}z_{i}\right)\right) \end{array}\right]\end{array} $$

(28)

The joint posterior distribution of all parameters is proportional to the product of the likelihood and the prior densities and is given by

$$ \begin{array}{rll} && f\left(\mbox{parameters}|\mbox{data}\right)\\ &&\propto\begin{array}{c} \displaystyle\prod\nolimits_{i=1}^{N}\left[\begin{array}{c} \displaystyle\prod\nolimits_{t=1}^{T_{i}}\left[\begin{array}{c} \left[|\Sigma|^{-0.5}\exp\left(-\frac{1}{2}\left(d_{it}-g_{it}\right)'\Sigma^{-1}\left(d_{it}-g_{it}\right)\right)\right]^{P_{it}}\\[10pt] \left[\exp\left(-\frac{1}{2}\left(P_{it}^{*}-Y_{it}\beta_{i}\right)^{2}\right)\right]^{1-P_{it}}\\[10pt] \left[1\left(P_{it}^{*}>0\right)\right]^{P_{it}}\left[1\left(P_{it}^{*}<0\right)\right]^{1-P_{it}} \end{array}\right]\\[12pt] |V_{\alpha}|^{-0.5}\exp\left(-\frac{1}{2}\left(\alpha_{i}-M_{\alpha}z_{i}\right)'V_{\alpha}^{-1}\left(\alpha_{i}-M_{\alpha}z_{i}\right)\right)\\[10pt] |V_{\beta}|^{-0.5}\exp\left(-\frac{1}{2}\left(\beta_{i}-M_{\beta}z_{i}\right)'V_{\beta}^{-1}\left(\beta_{i}-M_{\beta}z_{i}\right)\right) \end{array}\right]\\[10pt] |V_{\alpha}\otimes A^{-1}|^{-0.5}\exp\left(-\frac{1}{2}\left(m_{\alpha}-\bar{\alpha}\right)'\left(V_{\alpha}\otimes A^{-1}\right)^{-1}\left(m_{\alpha}-\bar{\alpha}\right)\right)\\[10pt] |V_{\beta}\otimes B^{-1}|^{-0.5}\exp\left(-\frac{1}{2}\left(m_{\beta}-\bar{\beta}\right)'\left(V_{\beta}\otimes B^{-1}\right)^{-1}\left(m_{\beta}-\bar{\beta}\right)\right)\\ |V_{\alpha}|^{-\frac{\mu_{\alpha}+k_{1}+1}{2}}\mbox{etr}\left(-\frac{1}{2}S_{\alpha}V_{\alpha}^{-1}\right)|V_{\beta}|^{-\frac{\mu_{\beta}+k_{1}+1}{2}}\mbox{etr}\left(-\frac{1}{2}S_{\beta}V_{\beta}^{-1}\right)\\[10pt] \left(\sigma^{2}\right)^{-\left(\frac{\nu_{0}}{2}+1\right)}\exp\left(-\frac{\nu_{0}s_{0}^{2}}{2\sigma^{2}}\right)\left(\frac{1+\rho}{2}\right)^{\lambda_{1}-1}\left(\frac{1-\rho}{2}\right)^{\lambda_{2}-1} \end{array}\\ \end{array} $$

(29)

Appendix B: Full conditional distributions

The full conditional distributions for each parameter vector is obtained by taking out all the terms from the joint posterior distribution in Eq. 29, since the other terms affect only the proportionality constant. We inspect these terms to see if they are from known distribution families. The joint posterior in our case is somewhat atypical because of the selection problem. It may appear that the full conditional distributions of α _i and β _i as well as those for $P_{it}^{*}$ are not from known distribution families if we just inspect them. However, on closer inspection, it turns out that they can be written as normal distributions.

In order to see this, it is useful to apply a simple trick of converting the joint density of d _it ¹⁹ into the product of conditional density of $\left(b_{it}|P_{it}^{*}\right)$ and the marginal density of $P_{it}^{*}$ when writing the full conditional density for α _i . Similarly, we write this joint density as the product of the conditional density of $\left(P_{it}^{*}|b_{it}\right)$ and the marginal density of b _it when writing the full conditional density for β _i .

For this purpose, it is useful to note that if

$$ \left(\begin{array}{c} \theta_{1}\\ \theta_{2} \end{array}\right)\sim N\left[\left(\begin{array}{c} \mu_{1}\\ \mu_{2} \end{array}\right),\Gamma=\left(\begin{array}{cc} \Gamma_{11} & \Gamma_{12}\\ \Gamma_{12} & \Gamma_{22} \end{array}\right)\right] $$

(30)

then

$$ \theta_{1}|\theta_{2}\sim N\left[\mu_{1}-V_{11}^{-1}V_{12}\left(\theta_{2}-\mu_{2}\right),V_{11}^{-1}\right]\mbox{ where }V=\left(\begin{array}{cc} V_{11} & V_{12}\\ V_{12} & V_{22} \end{array}\right)=\Gamma^{-1} $$

(31)

Note also that

$$ \Sigma^{-1}=\left(\begin{array}{cc} \frac{1}{\sigma^{2}\left(1-\rho^{2}\right)} & -\frac{\rho}{\sigma\left(1-\rho^{2}\right)}\\ \\ -\frac{\rho}{\sigma\left(1-\rho^{2}\right)} & \frac{1}{\left(1-\rho^{2}\right)} \end{array}\right) $$

(32)

Thus

$$ b_{it}|P_{it}^{*}\sim N\left[X_{it}\alpha_{i}+\sigma\rho\left(P_{it}^{*}-Y_{it}\beta_{i}\right),\sigma^{2}\left(1-\rho^{2}\right)\right] $$

(33)

and

$$ P_{it}^{*}|b_{it}\sim N\left[Y_{it}\beta_{i}+\frac{\rho}{\sigma}\left(b_{it}-X_{it}\alpha_{i}\right),1-\rho^{2}\right] $$

(34)

Thus, we can write the density of d _it (i.e. the joint density of b _it and $P_{it}^{*}$) in the first line of Eq. 29 in the following two ways

$$ f\left(d_{it}|\cdot\right)=f\left(b_{it}|P_{it}^{*},\cdot\right)f\left(P_{it}^{*}|\cdot\right) $$

(35)

and

$$ f\left(d_{it}|\cdot\right)=f\left(P_{it}^{*}|b_{it},\cdot\right)f\left(b_{it}|\cdot\right) $$

(36)

Then, the full conditional distributions of α _i and β _i start looking like the familiar normal distribution kernels. Thus, we have the following full conditional distributions for the parameters

• 1. α _i |·

  In order to derive the full conditional distribution of α _i , we first rewrite the joint posterior density in Eq. 29 using Eqs. 33 and 35 and then select all the terms that contain α _i . We can ignore the other terms since they affect only the proportionality constant and not the kernel of the density.

  $$ \begin{aligned} f\left(\alpha_{i}|\cdot\right)\propto\begin{array}{c} \displaystyle\prod\nolimits_{t=1}^{T_{i}}\left[\exp\left(-\frac{1}{2\sigma^{2}\left(1-\rho^{2}\right)}\left(b_{it}-X_{it}\alpha_{i}-\sigma\rho\left(P_{it}^{*}-Y_{it}\beta_{i}\right)\right)^{2}\right)\right]^{P_{it}}\\ \exp\left(-\frac{1}{2}\left(\alpha_{i}-M_{\alpha}z_{i}\right)'V_{\alpha}^{-1}\left(\alpha_{i}-M_{\alpha}z_{i}\right)\right) \end{array}\\ \end{aligned} $$

  (37)

  By rearranging terms in the above expression, we can show that

  $$ \begin{aligned}[b] \alpha_{i}|&\cdot\sim N\left[\tilde{\alpha}_{i},\left(\tilde{X}_{i}'\tilde{X}_{i}+V_{\alpha}^{-1}\right)^{-1}\right]\\ \tilde{\alpha}_{i}&=\left(\tilde{X}_{i}'\tilde{X}_{i}+V_{\alpha}^{-1}\right)^{-1}\left(\tilde{X}_{i}'\tilde{b}_{i}+V_{\alpha}^{-1}M_{\alpha}z_{i}\right) \end{aligned} $$

  (38)

  where $\tilde{b}_{i}$ is the vector obtained by stacking $\frac{\left[b_{it}-\sigma\rho\left(P_{it}^{*}-Y_{it}\beta_{i}\right)\right]}{\sqrt{\sigma^{2}\left(1-\rho^{2}\right)}}$ for all t for which P _it  = 1, and $\tilde{X}_{i}$ is the matrix obtained by stacking $\frac{X_{it}}{\sqrt{\sigma^{2}\left(1-\rho^{2}\right)}}$ for all t for which P _it  = 1.

• 2. β _i |·

  The derivation of the full conditional distribution of β _i is similar to that for α _i . However, there is one significant difference. In the case of α _i , the full conditional was affected only by those observations where P _it  = 1. In the case of the full conditional distribution of β _i , observations where P _i t = 0 also enter, but differently from those where P _it  = 0. Rewriting the joint posterior density in Eq. 29 using Eqs. 34 and 36 and selecting the terms that contain β _i gives us the following full conditional density

  $$ \begin{aligned} f\left(\beta_{i}|\cdot\right)\propto\begin{array}{c} \displaystyle\prod\nolimits_{t=1}^{T_{i}}\left\{\! \begin{array}{c} \left[\!\exp\left(-\frac{1}{2\left(1-\rho^{2}\right)}\left(P_{it}^{*}-Y_{it}\beta_{i}-\frac{\rho}{\sigma}\left(b_{it}-X_{it}\alpha_{i}\right)\right)^{2}\right)\!\right]^{P_{it}}\\[12pt] \left[\exp\left(-\frac{1}{2}\left(P_{it}^{*}-Y_{it}\beta_{i}\right)\right)\right]^{1-P_{it}} \end{array}\!\right\} \\[12pt] \exp\left(-\frac{1}{2}\left(\beta_{i}-M_{\beta}z_{i}\right)'V_{\beta}^{-1}\left(\beta_{i}-M_{\beta}z_{i}\right)\right) \end{array}\\ \end{aligned} $$

  (39)

  The full conditional distribution of β _i can be shown to be

  $$ \begin{aligned}[b] \beta_{i}|\cdot&\sim N\left[\tilde{\beta}_{i},\left(\tilde{Y}_{i}'\tilde{Y}_{i}+\ddot{Y}_{i}'\ddot{Y}_{i}+V_{\beta}^{-1}\right)^{-1}\right]\\ \tilde{\beta}_{i}&=\left(\tilde{Y}_{i}'\tilde{Y}_{i}+\ddot{Y}_{i}'\ddot{Y}_{i}+V_{\beta}^{-1}\right)^{-1}\left(\tilde{Y}_{i}'\tilde{P}_{i}^{*}+\ddot{Y}_{i}'\ddot{P}_{i}^{*}+V_{\beta}^{-1}M_{\beta}z_{i}\right) \end{aligned} $$

  (40)

  where $\tilde{P}_{i}^{*}$ is the vector obtained by stacking $\frac{\left[P_{it}^{*}-\frac{\rho}{\sigma}\left(b_{it}-X_{it}\alpha_{i}\right)\right]}{\sqrt{1-\rho^{2}}}$ for all t for which P _it  = 1, $\tilde{Y}_{i}$ is the matrix obtained by stacking $\frac{Y_{it}}{\sqrt{1-\rho^{2}}}$ for all t for which P _it  = 1, $\ddot{P}_{it}^{*}$ is the vector of stacked $P_{it}^{*}$ for all t for which P _it  = 0 and $\ddot{Y}_{it}$ is the matrix obtained by stacking Y _it for all t for which P _it  = 0.

• 3. $m_{\alpha}=\mbox{vec}\left(M_{\alpha}\right)|\cdot$ and $m_{\beta}=\mbox{vec}\left(M_{\beta}\right)|\cdot$

  $$ \begin{aligned}[b] m_{\alpha}|\cdot&\sim n\left[\tilde{m}_{\alpha},\left(Z'Z+A\right)^{-1}\right]\\ \tilde{m}_{\alpha}&=\mbox{vec}\left(\tilde{M}_{\alpha}\right)\mbox{ where }\tilde{M}_{\alpha}=\left(Z'Z+A\right)^{-1}\left(z'\alpha+A\bar{M}_{\alpha}\right) \end{aligned} $$

  (41)

  Z is the matrix formed by stacking z _i for all i, α is the matrix formed by stacking all α _i and $\bar{M}_{\alpha}$ is formed by stacking $\bar{\alpha}$ a column at a time.

  Similarly,

  $$ \begin{aligned}[b] m_{\beta}|\cdot&\sim N\left[\tilde{m}_{\beta},\left(Z'Z+B\right)^{-1}\right]\\ \tilde{m}_{\beta}&=\mbox{vec}\left(\tilde{M}_{\beta}\right)\mbox{ where }\tilde{M}_{\beta}=\left(Z'Z+B\right)^{-1}\left(Z'\beta+B\bar{M}_{\beta}\right) \end{aligned} $$

  (42)

  Z is the matrix formed by stacking z _i for all i, β is the matrix formed by stacking all β _i and $\bar{M}_{\beta}$ is formed by stacking $\bar{\beta}$ a column at a time.

• 4. V _α |· and V _β |·

  $$ \begin{aligned}[b] V_{\alpha}|\cdot&\sim\mbox{Inverse Wishart}\left(\mu_{\alpha}+N,S_{\alpha}+E_{\alpha}'E_{\alpha}\right)\\ E_{\alpha}&=\alpha-Z\tilde{M}_{\alpha}+\left(\tilde{M}_{\alpha}-\bar{M}_{\alpha}\right)^{-1}A\left(\tilde{M}_{\alpha}-\bar{M}_{\alpha}\right) \end{aligned} $$

  (43)

  $\tilde{M}_{\alpha}$ and $\bar{M}_{\alpha}$ have already been defined in Eq. 41 and N is the total number of consumers.

  Similarly,

  $$ \begin{aligned}[b] V_{\beta}|\cdot&\sim\mbox{Inverse Wishart}\left(\mu_{\beta}+N,S_{\beta}+E_{\beta}'E_{\beta}\right)\\ E_{\beta}&=\beta-Z\tilde{M}_{\beta}+\left(\tilde{M}_{\beta}-\bar{M}_{\beta}\right)^{-1}B\left(\tilde{M}_{\beta}-\bar{M}_{\beta}\right) \end{aligned} $$

  (44)

• 5. $P_{it}^{*}|\cdot$

  Using data augmentation (Tanner and Wong 1987), we can treat the vector of $P_{it}^{*}$ as parameters and make draws for them from their full conditional distribution like in the case of the other parameters Note that when P _it  = 1, $P_{it}^{*}$ is drawn from a left-truncated normal distribution, while it is drawn from a right-truncated normal distribution when P _it  = 0. Where our specific formulation differs from the standard binomial probit model is that the since we have a bivariate distribution of the errors, there are further restrictions placed on $P_{it}^{*}$.

  $$ P_{it}^{*}|\cdot\sim\begin{array}{cc} N\left[Y_{it}\beta_{i}+\frac{\rho}{\sigma}\left(b_{it}-X_{it}\alpha_{i}\right),1-\rho^{2}\right]1\left(P_{it}^{*}>0\right) & \mbox{ when }P_{it}=1\\ \\ N\left[Y_{it}\beta_{i},1\right]1\left(P_{it}^{*}<0\right) & \mbox{ when }P_{it}=0 \end{array} $$

  (45)

• 6. σ ²|· and ρ|·

  The full conditional distributions of these two parameters are not from known distribution families and hence we cannot draw from them directly. We use the Random Walk Metropolis Hastings algorithm (Chib and Greenberg 1995) to make draws from these distributions. We use normal candidate densities for this purpose, using standard errors from likelihood estimates of these parameters and a scaling/tuning parameter that was set by maximizing the relative numerical efficiency to fix the variances of these candidate densities. More details are available from the authors on request.