Round 10 of Grand Prix Providence featured a classic moment. The match was between Pro Tour Venice 2003 champion Osyp Lebedowicz and Hall-of-Famer Owen Turtenwald, two of the game’s all-time greats. In Game 2, Osyp got on the front foot with a 1/3 Wildgrowth Walker on turn 2 and a 1/4 Atzocan Archer on turn 3. Owen then cast a powerful uncommon: Shapers of Nature.

Although the 3/3 was the biggest creature on the battlefield, Osyp knows how to drive his Humans to school. He played his fourth land and turned his Atzocan Archer sideways, representing Crash the Ramparts or Sure Strike. Owen surely knew that it could be a bluff, but he also didn’t want to lose his premium uncommon to a simple combat trick. Given that this was a team event, he conferred with his teammate and 2017 World Champion, William “Huey” Jensen. We don’t have audio, so I don’t know what they said exactly, but here’s my best guess at the conversation that might have taken place:

“Huey, he’s attacking his Archer into my Shapers of Nature. What would you do here?”

“Just block. If he had a trick, he would’ve attacked with his Wildgrowth Walker as well.”

“Hmm, I don’t know. It could be a ruse. I don’t want to lose my Shapers of Nature.”

“Fair. Just do what you feel is right. Either play is fine.”

After some additional thought, Owen decided to take the damage.

In classic Osyp fashion, a completely over-the-top celebration ensued, as all he had in hand was a bunch of creatures and a ham sandwich. Given that these two players know each other pretty well and that Owen had taken some time to make his decision, these rub-ins were pretty funny. I can only imagine Osyp shouting, “Best player in the world? Just bluffed you for a free point of damage. Yeah!”

It turned the situation into a comedic moment that put a smile on the face of everyone at the table, including Owen.

Every point counts Frank

— Osyp Lebedowicz (@OsypL) October 1, 2017

But the situation was also interesting for its strategic aspects. Today, I’ll apply concepts from non-cooperative game theory (a branch of mathematics that deals with the modeling and analysis of interactive strategic behavior) to provide a framework for thinking about these bluffing scenarios.

### Did Osyp and Owen make the right decisions?

To analyze that, let’s try to make a mathematical model of the situation. I’ll cast the decisions and outcomes into the framework of a non-cooperative game, which features three elements: players, strategies, and payoffs.

A strategy for a player should describe a choice for every possible scenario (or, in more technical terms, for every information set) that this player can distinguish. For Osyp, there are two different scenarios that he might encounter at the start of his turn with this board state: He is either holding a trick or he is not holding a trick. In both cases, he can choose between (1) attacking with both creatures, (2) attacking with Atzocan Archer, or (3) not attacking at all. Technically, he could also attack with the Wildgrowth Walker only, but that would be a rather strange play that I don’t immediately see a benefit for, so to obtain a sufficiently simple model I excluded that option. Given that Osyp has two scenarios with three options each, he has 3^2=9 complete strategies. For example, “attack with both creatures if I hold a trick, no attacks if I lack a trick” would be one of those 9.

For Owen, there are two different scenarios in which he has to make a decision: He is either being attacked by both creatures or attacked by Atzocan Archer only. In both cases, he can choose between (1) blocking the smallest attacker or (2) not blocking. Technically, he could also block the 1/4 if he’s attacked by both the 1/3 and the 1/4, but again I see no benefit to blocking the 1/4 over the 1/3, so I excluded that option as well. Hence, Owen has two scenarios with two options each, which results in 2^2=4 complete strategies. As an example, “block if being attacked by the 1/4 only, no blocks if being attacked by both” would be one of those 4.

Next, I have to specify expected payoffs for every outcome. An outcome is the combination of a strategy for each player. I’ll represent the payoffs in terms of the probability that Osyp will win the game, under the assumption that Osyp’s win probability at the start of the turn is 50%. Estimating those probabilities is tough and arguably the hardest part of this exercise. To keep my sanity, I’ll simplify considerably, looking only 1 turn cycle ahead (thereby ignoring the value that a trick might have in subsequent turns) and base the probabilities on the following.

- If Osyp manages to deal 1 damage to Owen, then this increases his chances of winning by 1% (i.e., from 50% to 51%)
- If Osyp manages to kill Shapers of Nature with a trick, then this increases his chances of winning by 8%
- If Osyp loses his 1/3 in a chump-attack, then this decreases his chances of winning by 4%
- If Osyp leaves his 1/4 behind as a blocker, then this increases his chances of winning by 0.5%.

That last number is based on the notion that an untapped 1/4 can prevent 3 damage if Owen misses his fourth land drop. Osyp surely won’t block an attacking Shapers of Nature if Owen has the mana to activate its ability, but if Owen misses his fourth land drop (which, given that he hit his third one, will happen in about 15% of the time), then an untapped 1/4 will be useful. So there is a benefit to keeping Atzocan Archer behind—it wasn’t a completely free attack!

Finally, I will assume that Osyp is granted a trick in hand with 25% probability. This is indeed the number on turn 4 for a deck with 1 copy of Sure Strike or Crash the Ramparts. A single trick seems typical for a 40-card red-green deck, and I will assume that both players know this. I also assume that both players are know and agree with the above-described ways of how Osyp’s win percentage can change.

With all ingredients in place, I can now represent the game as follows. The rows correspond to Osyp’s strategies, the columns represent Owen’s strategies, and the number for every outcome is the expected probability that Osyp will win the game.

If attacked by both |
Block 1/3 | Block 1/3 | No blocks | No blocks | ||

If attacked by 1/4 |
Block 1/4 | No blocks | Block 1/4 | No blocks | ||

Trick |
No trick |
|||||

Attack with 1/4 | Attack with 1/4 | 52.00% | 51.00% | 52.00% | 51.00% | |

Attack with 1/4 | Attack with both | 49.75% | 48.00% | 53.50% | 51.75% | |

Attack with 1/4 | No attacks | 52.38% | 50.63% | 52.38% | 50.63% | |

Attack with both | Attack with 1/4 | 52.25% | 53.00% | 50.50% | 51.25% | |

Attack with both | Attack with both | 50.00% | 50.00% | 52.00% | 52.00% | |

Attack with both | No attacks | 52.63% | 52.63% | 50.88% | 50.88% | |

No attacks | Attack with 1/4 | 50.13% | 50.88% | 50.13% | 50.88% | |

No attacks | Attack with both | 47.88% | 47.88% | 51.63% | 51.63% | |

No attacks | No attacks | 50.50% | 50.50% | 50.50% | 50.50% |

To illustrate how these numbers arose from my assumptions, consider as an example the outcome where Osyp attacks with both creatures if he has a trick and doesn’t attack if he has no trick (the sixth row) while Owen blocks the 1/3 if he’s attacked by both creatures and blocks the 1/4 if he’s attacked by only the 1/4 (the first column). Irrespective of strategy, Osyp is granted a trick once every 4 games, and in that case we’ll see a dead Shapers of Nature plus 1 point of damage, which corresponds to a probability of 50%+8%+1%=59% that Osyp will win the game. In the 3 out of every 4 games where Osyp wouldn’t have a trick, he makes no attacks, which corresponds to 50.5%. Finally, 0.25 * 59% + 0.75 * 50.5% = 52.63%. You may interpret this number as the fraction of games won by Osyp if both players prescribe their full strategies in advance, Osyp is randomly granted a trick in 1/4th of the games, and the subsequent decisions are made according to the prescribed strategies.

As you can imagine from the calculation underlying the 52.63% number—which is only one out of 36—no human could compile this matrix in any reasonable amount of time. What’s more, it’s based on a bunch of assumptions and approximations—imagine if I wanted to make it more accurate by considering more than one turn cycle ahead or by enriching the game model with more information sets concerning each other’s hands. It would become way too difficult! Yet, even though this model is a big abstraction of reality, it does give an illuminating framework that can help guide our conceptual thinking about the game.

As a final note, the strategies from the final three rows are all dominated—there is no reason to choose them over the corresponding strategy that does attack when you hold a trick. They are simply included for completeness.

### So what are the optimal decisions?

Well, that depends on the opponent’s strategy.

Let’s put ourselves in Osyp’s shoes. If you expect that Owen will never block—the far right column—then your best response is to attack with both creatures whether you have a trick or not. Indeed, this way you get 52%, which is the maximum of all values in the far right column.

But if you expect that Owen will always block—the far left column—then your best response is to attack with both if you have a trick and to not attack if you don’t. This way, you get the maximum from that column: 52.63%.

Flipping the tables and putting ourselves in Owen’s shoes, you want to base your plays on your estimation of Osyp’s strategy as well. If, for instance, you believe that he will attack with both if he has a trick and will attack with only the 1/4 if he doesn’t—the fourth row—then the best response is to not block if attacked by both and to block the 1/4 if only the 1/4 is attacking. This gives Osyp the minimum value from that row: 50.5%.

We can keep going like this, but the conclusion would be that there is no strategy that dominates all others for any player. So, there is no “optimal” play in this context. Moreover, there is no mutual best response, and hence no pure-strategy Nash equilibrium.

### An optimal play doesn’t exist for this setting

Many players cling to a myth that there is always one technically correct play for every situation in Magic, but if the outcomes depend on your opponent’s strategy, then there is no decision that is better than all of the others.

You can only talk about an optimal play if you fix your opponent’s strategy. Then, it reduces to a perfect-information game, like chess, where such technically correct plays do exist. And to be fair—this is usually how you would tackle decisions like these, at least subconsciously: You make your best guess as to how your opponent will be playing (hopefully in a way that can be exploited) and then you figure out your best response to that. But these guesses would be based on beliefs or estimates, not merely on the nature of the game matrix itself. And if you guess wrongly, then your opponent could exploit that.

Alternatively, you could take a pessimist’s view and say that the best choice for Osyp is the one that gives the best outcome if Owen always has the soul read on your strategy. In this case, Osyp should always attack with his 1/4 only (the first row) because then he’s guaranteed at least a 51% to win the game—more than the worst-case scenario for any of his other strategies. But as you’ll see later, you can get more by mixing your plays.

Indeed, unless you are completely sure that, say, your opponent literally always blocks in spots like these, then the closest you can get to something optimal is an “optimal” (meaning equilibrium) bluffing frequency.

### Mixing your plays can pay off

Mixing your plays means picking a probability distribution over your strategies. Basically, Owen might say, “all right Osyp, if you attack, then I’ll roll a six-sided dice and block if I roll 1 or 2— otherwise I’ll take the damage.”

The key here is to choose the probabilities such that the opponent becomes indifferent between their strategies. A pair of two mixed strategies where no player has an incentive to deviate is called a mixed-strategy Nash equilibrium.

For this game, there are multiple such equilibria. In one, Osyp plays (0, 2/7, 0, 2/3, 1/21, 0, 0, 0, 0), i.e., row 2 with probability 2/7, row 4 with probability 2/3, and row 5, while Owen plays (0, 2/7, 3/7, 2/7). Another equilibrium is where Osyp plays (2/7, 0, 0, 8/21, 1/3, 0, 0, 0, 0) while Owen plays (2/7, 0, 1/7, 4/7). There are others as well.

Yet, both above-described equilibria come down to the following:

- If Osyp has a trick, he attacks with only Atzocan Archer with probability 2/7 and attacks with both creatures with probability 5/7
- If Osyp doesn’t have a trick, he attacks with only Atzocan Archer with probability 2/3 and attacks with both creatures with probability 1/3.
- If Owen is attacked by both creatures, he blocks Wildgrowth Walker with probability 2/7 and doesn’t block with probability 5/7
- If Owen is attacked by only Atzocan Archer, he blocks with probability 3/7 and doesn’t block with probability 4/7

It is easy to verify that if both players randomly choose their plays according to the above-described probabilities, then Osyp has a 51.43% probability to win the game. Note that this is more than 51%, which was the best that a pessimist with pure strategies could guarantee. What’s more, equilibrium strategies cannot be exploited—no matter what Owen might do, Osyp’s win percentage would always be the same, and vice versa.

### Wait, but shouldn’t Owen have picked up on the fact that the 1/3 stayed back?

That is a good question to ask. The line of reasoning would go as follows: If Osyp had the trick, then the 1/3 could have safely attacked as well. Given that it didn’t, Owen should have deduced that Osyp was bluffing. In terms of our game model, Owen should be playing the third column.

Then again, if Osyp knows that Owen would think that way, he could go one level deeper: Attack with only the 1/4 if have the trick, and attack with both if you don’t have it. In other words, play the second row. I love making attacks like these when I’m actually holding the combat trick.

Indeed, such strategies are represented in the mixed equilibrium. Intuitively, there are times where throwing your opponent off is worthwhile, and that make it so the contents of your hand can’t be predicted with certainty from the plays that you make. So even though not attacking with the 1/3 might signal something, it doesn’t guarantee that Osyp has no trick, and Owen should definitely not always block.

### Should you celebrate after a successful bluff?

Osyp’s celebration, while funny, could have given away that he didn’t have the combat trick in hand. But that’s only true if he would only celebrate after a successful bluff.

If he also celebrates with the exact same frequency after such a combat step where he actually had the trick in hand, then Owen could gain no information from the act of celebration. So that’s the key—celebrate just as often and just as enthusiastically in all scenarios.

That said, the whole situation would have been an instant classic if Osyp had attacked Owen on the next turn, got a block, and revealed that he had a trick in hand all along. This didn’t happen here, but I have hopes for the future.

### Embrace the Dice Man

In situations like the one that Osyp and Owen found themselves in, you should sometimes bluff and sometimes call. If you do the same every time, then your play style may become known, and astute opponents may take advantage of that. But if you randomize your plays with the right frequencies, then you don’t become exploitable yourself, and you can win more games than you would if you would always play in a certain fixed way.

So next time your opponent attacks your 2/2 into your 4/4, just roll a dice. Tell your opponent “I’ll block if it lands on a 1 or 2” (or pick another probability based on the situation at hand) and do whatever the dice tells you to. I have done this on numerous occasions on the Pro Tour myself. The puzzled look on your opponent’s face will be priceless, it brings some additional excitement, and you may maximize your long-term win percentage if you might face your opponent in a similar situation again.