Arbitrage challenge

[Justin7]

You have one sportsbook that lets you make 3-team 6-point teasers at +180. They have spread blow (1). Assume you can also bet any moneyline at Pinnacle (2) or from (1). Finally, assume you can also get +3000 on "Will this team win by exactly 1", and can also bet "Will this team win by exactly 2" at +3000 at matchbook.

You have exactly $1000 you can risk, however you want among these books. What is the most you can GUARANTEE via arbitrage, if none of these odds change?

(1) from Bookmaker

301 PITTSBURGH -7½-110 o44-110 -360
302 ST. LOUIS +7½-110 u44-110 +300
NFL First Half Lines - Dec 20
5:15 PM12 /20 /07
337 NY JETS +8½-110 o37½-110 +330
338 TENNESSEE -8½-110 u37½-110 -400
1:05 PM12 /23 /07
339 TAMPA BAY -7-110 o36½-110 -300
340 SAN FRANCISCO +7-110 u36½-110 +250
NFL - Dec 24
NFL
MONDAY, DECEMBER 24TH
5:00 PM12 /24 /07
# Spread Total Money Line
343 DENVER +8-110 o47½-110 +315
344 SAN DIEGO -8-110 u47½-110 -375

(2) from Pinnacle
Sun 12/23 337 New York Jets +9 -112 +335 OVER 37 -106
01:15 PM 338 Tennessee Titans -9 +104 -365 UNDER 37 -104

Sun 12/23 339 Tampa Bay Buccaneers -7 +102 -300 OVER 36.5 -109
01:05 PM 340 San Francisco 49ers +7 -110 +270 UNDER 36.5 -101

[RickySteve]

What price do the teasers reduce to? How do they handle 2-team ties?

[RickySteve]

You also need a prop on a tie.

[Justin7]

2-teamers reduce to -110 (retail). There is NO prop on tie.

[RickySteve]

Without a prop on a tie, you can't guarantee a payout.

[Justin7]

Sorry, I've been out of practice. Ignore the tie (you have inside info that the tie will hit 0%).

I nearly put in a caveat... RickySteve and Ganchrow, let everyone else try for 24 hours. But you're on top of it, fire away.

[RickySteve]

I defer to the right honorable gentleman from Forest Hills.

[Justin7]

So minimum was a $193 loss?

I would note that 3 teasable games are on 3 different days. There should be a +EV arbitrage.

[Thremp]

I would note that 3 teasable games are on 3 different days.

This makes no sense to me. Why would the time period change the EV?

[rm18]

This makes no sense to me. Why would the time period change the EV?

guaranteed profit instead of positive expectancy

[Thremp]

guaranteed profit instead of positive expectancy

Explain.

[Ganchrow]

So after conferring with Professor RickySteve we came up with the following:

If the only goal were zero variance then (assuming no tie games) by taking each of the following teasers and hedging as appropriate you'd net the following guaranteed returns (as a percentage of the teaser wager):

PIT-TB-SD => 3.55%
PIT-TEN-SD => 1.48%

If the NYJ vs. TEN and TB vs. SF were not contemporaneous then you'd also be able to choose from:

PIT-TEN-TB => 1.93%
TEN-TB-SD => -1.48%

But that's not the case and so that's irrelevant to this question.

So the highest possible risk-free return would be 3.55% of the notional teaser value. This would be achieved by taking the PIT-TB-SD teaser and hedging as appropriate.

If limited to a $1,000 bankroll then the highest possible zero-variance return using the PIT-TB-SD teaser would be $12.83 (a $361.73 bet on the teaser).

In keeping with Justin's wishes I won't divulge the hedging strategy that produces these results unless he gives the go-ahead. Feel free to PM if you need the answer now.

Of course the more interesting (and labor-intensive) problem would involve determining the Kelly-optimal allocation given these available bets and some assumed W/L/P probabilities.

[Thremp]

Explain.

I think I see it now after Ganch's post.

[Ganchrow]

Since no one's answered and Justin's 24-hour period's expired:

PIT/TB/SD Teaser

Initial Bets:
<style> .GANCH { font-size: 12px; font-family: Verdana, Arial, Helvetica, sans-serif; }</style> <table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> TEASER +180 </td> <td align=right class=GANCH> $361.73
</td> </tr> <tr> <td align=left class=GANCH> STL +300 </td> <td align=right class=GANCH> $130.47
</td> </tr> <tr> <td align=left class=GANCH> PIT BY 1 +3000 </td> <td align=right class=GANCH> $16.83
</td> </tr> </table>
Only If PIT by > 1:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> SF +270 </td> <td align=right class=GANCH> $196.08
</td> </tr> <tr> <td align=left class=GANCH> TB BY 1 +3000 </td> <td align=right class=GANCH> $7.56
</td> </tr> </table>
Only If PIT by > 1 AND If TB by > 1:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> DEN +315 </td> <td align=right class=GANCH> $244.06
</td> </tr> <tr> <td align=left class=GANCH> SD BY 1 +3000 </td> <td align=right class=GANCH> $32.67
</td> </tr> <tr> <td align=left class=GANCH> SD BY 2 +2940 </td> <td align=right class=GANCH> $10.60
</td> </tr> </table>
Only If PIT by > 1 AND If TB by exactly 1:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> DEN +315 </td> <td align=right class=GANCH> $166.40
</td> </tr> <tr> <td align=left class=GANCH> SD BY 1 +3000 </td> <td align=right class=GANCH> $22.28
</td> </tr> <tr> <td align=left class=GANCH> SD BY 2 +2940 </td> <td align=right class=GANCH> $10.82
</td> </tr> </table>

In this manner the player will risk at most $1,000 (only if both PIT and TB cover the teased spreads) and will, regardless of outcome (save a tied game), have guaranteed himself a $12.83 profit (that's 3.55% of the $361.73 notional value of the teaser).

<hr>
(The PIT/TEN/SD teaser is inferior to PIT/TB/SD teaser as far maximum zero-variance profitability goes, but for the sake of completeness, however, the solution follows.)

PIT/TEN/SD Teaser

Initial Bets:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> TEASER +180 </td> <td align=right class=GANCH> $359.04
</td> </tr> <tr> <td align=left class=GANCH> STL +300 </td> <td align=right class=GANCH> $126.92
</td> </tr> <tr> <td align=left class=GANCH> PIT BY 1 +3000 </td> <td align=right class=GANCH> $16.38
</td> </tr> </table>
Only If PIT by > 1:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> NYJ +335 </td> <td align=right class=GANCH> $165.55
</td> </tr> <tr> <td align=left class=GANCH> TENN BY 1 +3000 </td> <td align=right class=GANCH> $23.23
</td> </tr> <tr> <td align=left class=GANCH> TENN BY 2 +2940 </td> <td align=right class=GANCH> $23.69
</td> </tr> </table>
Only If PIT by > 1 AND If TENN by > 1:
<table border=0 cellpadding=1 cellspacing=2> <tr> <td align=left class=GANCH> DEN +315 </td> <td align=right class=GANCH> $242.25
</td> </tr> <tr> <td align=left class=GANCH> SD BY 1 +3000 </td> <td align=right class=GANCH> $32.43
</td> </tr> <tr> <td align=left class=GANCH> SD BY 2 +2940 </td> <td align=right class=GANCH> $10.52
</td> </tr> </table>

In this manner the player will risk at most $1,000 (only if both PIT and TENN cover the teased spreads) and will, regardless of outcome (save a tied game), have guaranteed himself a $5.33 profit (that's 1.48% of the $359.04 notional value of the teaser).

[Justin7]

Nice Ganch.

Who says we never give away free money here?