<p>Saying 'rational' does not tell you anything about an agent's goal function, just that given a goal function the agent is doing their best to maximize its value.</p>
<p>In your case an agent who has a goal function of maximizing the probability of winning &lt;sufficiently large amount of money&gt; will put all their money towards the lottery if that offers the best chance of winning &lt;sufficiently large amount of money&gt;.</p>
<hr />
<p>This is IMO a big problem in rationality based modelling, since all sorts of behavior can be considered rational, many things are 'rationalizable' given the proper goal function. Thus the choice of goal function supporting a theoretical model is frequently arbitrary and little effort is made at validating it.</p>

CC BY-SA 4.0
2023-07-25T19:34:04.177Z
2023-07-25T19:34:04.177Z
Giskard
<h4>Daily Keno's too negative Expected Value looks scammy.</h4>
<p>For all the different wagers of <a href="https://www.olg.ca/en/lottery/play-daily-keno-encore/past-results.html#odds-and-payouts" rel="nofollow noreferrer">Keno's $10 PICK</a>, the odds of winning jackpot are selfsame : 1 in 2,147181. I computed their EV.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: center;">Wager</th>
<th style="text-align: center;">Jackpot</th>
<th style="text-align: center;">EV  = [(Jackpot — Wager)  × Probability] — Wager</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center;">\$10</td>
<td style="text-align: center;">\$2.5m</td>
<td style="text-align: center;"><span class="math-container">$\frac{2.5E6 - 10}{2147181} -10=-8.84 $</span></td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">1.25m</td>
<td style="text-align: center;"><span class="math-container">$-4.42$</span></td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">500k</td>
<td style="text-align: center;"><span class="math-container">$-1.77$</span></td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">250k</td>
<td style="text-align: center;"><span class="math-container">$-0.88$</span></td>
</tr>
</tbody>
</table>
</div>
<p>As many  play the same lottery repeatedly, I shall consider <a href="https://math.stackexchange.com/a/4685078"><span class="math-container">$\Pr($</span> you win jackpot at least once│<em>n</em>  plays) <span class="math-container">$= 1 - (1 - p)^n$</span></a>.</p>
<h4>But some rational players can sensibly tolerate Keno’s fairish <span class="math-container">$\Pr($</span> you win jackpot at least once│<em>n</em> plays).</h4>
<p>Before COVID, I spent \$5K USD on leisure travel. But I hanker to, and can, retire on \$1.25M. Then I can travel less, and spend \$3650 CAD/year (e.g. \$5/play × 2 plays/day  × 365 days/year) buying the 10 PICK \$5 Keno, <em><strong>twice daily.</strong></em> I prefer Keno's teensy chance of winning jackpot, over traveling's <span class="math-container">$\rightarrow 0^{+}$</span> chance  — assuming that I don't meet a multimillionaire during travel, and marry him!</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: center;">n</th>
<th style="text-align: center;">n/2 = days</th>
<th style="text-align: center;"><span class="math-container">$1 - (1 - \dfrac1{2147181})^n$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center;">21</td>
<td style="text-align: center;">10.5</td>
<td style="text-align: center;">= 0.00010 ≈ 1/97,599</td>
</tr>
<tr>
<td style="text-align: center;">215</td>
<td style="text-align: center;">107.5 <a href="https://planetcalc.com/7933/" rel="nofollow noreferrer">(= 3 months, 17 days)</a></td>
<td style="text-align: center;">= 0.00010 ≈ 1/10k</td>
</tr>
<tr>
<td style="text-align: center;">366</td>
<td style="text-align: center;">183 (= half a year)</td>
<td style="text-align: center;">= 0.00017 ≈ 1/5883</td>
</tr>
<tr>
<td style="text-align: center;">730</td>
<td style="text-align: center;">365 (= 1 year)</td>
<td style="text-align: center;">= 0.00034 ≈ 1/2941</td>
</tr>
<tr>
<td style="text-align: center;">1460</td>
<td style="text-align: center;">730 (= 2 years)</td>
<td style="text-align: center;">= 0.00068 ≈ 1/1471</td>
</tr>
<tr>
<td style="text-align: center;">2150</td>
<td style="text-align: center;">1075 (= 2 years, 11 months)</td>
<td style="text-align: center;">= 0.0010 ≈ 1/1000</td>
</tr>
</tbody>
</table>
</div>
<p>Some <em>Homo Economicus</em> can logically accept these humdrum probabilities, like <span class="math-container">$1/5883$</span> probability of winning $1.25M over 6 months.</p>
<blockquote>
<p><a href="https://money.stackexchange.com/a/106336">Playing the lottery can be worth it, even with negative expected value.</a></p>
</blockquote>
<blockquote>
<p><a href="https://money.stackexchange.com/a/63930">From a mathematical expected-value standpoint, there is no difference between gambling (e.g. buying a lottery ticket) and investing (e.g. buying a share of stock).</a></p>
</blockquote>
<h4>How can rational players decide between Keno’s EV and <span class="math-container">$\Pr($</span>you win jackpot at least once│<em>n</em> plays)?</h4>

<uncertainty>
How to decide whether to buy a lottery with a too negative EV, but passable $\Pr($you win jackpot at least once│n plays)?
Justice Blueprint
# How to decide whether to buy a lottery with a too negative EV, but passable $\Pr($you win jackpot at least once│n plays)?

(Asked by: username_0 on 2023-07-25)

#### Daily Keno's too negative Expected Value looks scammy.

For all the different wagers of [Keno's $10 PICK](https://www.olg.ca/en/lottery/play-daily-keno-encore/past-results.html#odds-and-payouts), the odds of winning jackpot are selfsame : 1 in 2,147181. I computed their EV.

| Wager | Jackpot | EV = [(Jackpot — Wager) × Probability] — Wager |
| --- | --- | --- |
| \$10 | \$2.5m | $\frac{2.5E6 - 10}{2147181} -10=-8.84 $ |
| 5 | 1.25m | $-4.42$ |
| 2 | 500k | $-1.77$ |
| 1 | 250k | $-0.88$ |

As many play the same lottery repeatedly, I shall consider [$\Pr($ you win jackpot at least once│*n* plays) $= 1 - (1 - p)^n$](https://math.stackexchange.com/a/4685078).

#### But some rational players can sensibly tolerate Keno’s fairish $\Pr($ you win jackpot at least once│*n* plays).

Before COVID, I spent \$5K USD on leisure travel. But I hanker to, and can, retire on \$1.25M. Then I can travel less, and spend \$3650 CAD/year (e.g. \$5/play × 2 plays/day × 365 days/year) buying the 10 PICK \$5 Keno, ***twice daily.*** I prefer Keno's teensy chance of winning jackpot, over traveling's $\rightarrow 0^{+}$ chance — assuming that I don't meet a multimillionaire during travel, and marry him!

| n | n/2 = days | $1 - (1 - \dfrac1{2147181})^n$ |
| --- | --- | --- |
| 21 | 10.5 | = 0.00010 ≈ 1/97,599 |
| 215 | 107.5 [(= 3 months, 17 days)](https://planetcalc.com/7933/) | = 0.00010 ≈ 1/10k |
| 366 | 183 (= half a year) | = 0.00017 ≈ 1/5883 |
| 730 | 365 (= 1 year) | = 0.00034 ≈ 1/2941 |
| 1460 | 730 (= 2 years) | = 0.00068 ≈ 1/1471 |
| 2150 | 1075 (= 2 years, 11 months) | = 0.0010 ≈ 1/1000 |

Some *Homo Economicus* can logically accept these humdrum probabilities, like $1/5883$ probability of winning $1.25M over 6 months.

> [Playing the lottery can be worth it, even with negative expected value.](https://money.stackexchange.com/a/106336)

> [From a mathematical expected-value standpoint, there is no difference between gambling (e.g. buying a lottery ticket) and investing (e.g. buying a share of stock).](https://money.stackexchange.com/a/63930)

#### How can rational players decide between Keno’s EV and $\Pr($you win jackpot at least once│*n* plays)?

**Tags:** uncertainty


---

## 1 answers

### Answer by username_1 on 2023-07-25. Score: 1 | Accepted answer

Saying 'rational' does not tell you anything about an agent's goal function, just that given a goal function the agent is doing their best to maximize its value.

In your case an agent who has a goal function of maximizing the probability of winning <sufficiently large amount of money> will put all their money towards the lottery if that offers the best chance of winning <sufficiently large amount of money>.

---

This is IMO a big problem in rationality based modelling, since all sorts of behavior can be considered rational, many things are 'rationalizable' given the proper goal function. Thus the choice of goal function supporting a theoretical model is frequently arbitrary and little effort is made at validating it.


---

