How often should you kick? Optimal kick probability in World of Warcraft PvP

TLDR: From the perspective of game theory, high impact kicks should be thrown less frequently than low impact kicks. This is because when a kick is impactful, there is a bigger incentive for your opponent to juke your kick. For lower impact kicks, you should kick more often (if you are going to kick at all) since your opponent gets a low payoff from juking. For most situations, a reasonable kick probability is around 30-40% of casts.

In WoW PvP, kicking and juking (fake-casting) has been common practice since arenas were introduced. These days, not only is juking common practice at all ratings, but holding one's kick is a deliberate part of many players' strategies (i.e. refraining from kicking with the expectation that the opponent will attempt to juke). But how often should we actually kick? We probably agree that 100% of the time is clearly too often, and 0% of the time is not often enough. But is there such a thing as an optimal kick probability?

How do we even begin answering this type of question? Fortunately, if we make some simplifying assumptions, we can reduce kicking and juking down to a game. In this context, I don't mean game as in a video game, but rather in the way that game theorists use the term. Within game theory, games are a set of actions that have well-defined effects. Rock-paper-scissors is an example of a game where two people make a choice that can each win, draw, or lose against the opponent's choice. If player 1 chooses rock while player 2 chooses scissors, player 1 wins the game. We can represent this game with the following table.

P1 Rock P1 Paper P1 Scissors
P2 Rock Draw P1 wins P2 wins
P2 Paper P2 wins Draw P1 wins
P2 Scissors P1 wins P2 wins Draw

The above is known as a payoff matrix and is a very common way of representing games. So what is the optimal strategy for a game like rock-paper-scissors? It depends on how you define optimal, but if you know that your opponent favours any one of the options (e.g. paper), the strategy which maximises your expected win rate is to choose the alternative that beats their preferred option (i.e. if they prefer paper, you should always choose scissors). Such strategies are known as exploitative strategies. They are called exploitative because they exploit weaknesses in your opponent's strategy. However, if you are playing repeated games of rock-paper-scissors, your opponent will likely pick up on your preference for scissors and adjust their strategy accordingly. You now enter into a meta-game where you're trying to get into each other's heads to figure out what you'll play next. These sorts of mindgames have given rise to some pretty special moments, such as this poker hand between Phil Ivey and Paul Jackson where neither player has anything, but are also certain that the other person has nothing. The result is that both players engage in spectacular high-stakes bluffs and counter-bluffs.

An alternative to trying to get into your opponent's head is to simply choose a strategy that cannot be exploited. In the poker world, this is often known as a "Game Theory Optimal" strategy (GTO for short). In game theory, it is known as a Nash equilibrium (named after the Nobel laureate John Nash, the protagonist of "A Beautiful Mind"). So what is the Nash equilibrium (or the GTO strategy) for rock-paper-scissors? In other words, what is the strategy which can never be exploited? It is simply to play each option with equal probability (33.33%). Note that this is not the strategy that will make you win the most, but the strategy that will guarantee that your opponent cannot adjust their strategy to exploit you. No matter what your opponent does, you will always win half of the time.

Generalising this to World of Warcraft, we might be able to come up with a kick probability that will always do well against any opponent. Why would you want to play a strategy that is game theory optimal? Surely the purpose of picking a strategy is to maximise your win rate? It's worth being clear that if you know that someone never jukes, you should just always kick their cast (provided it's worth kicking to begin with). However, the key thing to understand about exploitative strategies is that they themselves are exploitative. Thus, you run the risk of actually losing out by either choosing a suboptimal strategy, or getting exploited by someone else picking up on what you're doing. If you're facing moderately skillful opposition they will usually be fairly sensitive to how often you kick and adjust their gameplay accordingly.

So how can we work out the GTO kick frequency for WoW? The first step is to simplify kicking and juking to a situation where one player is either going to kick or not, and the other player is either going to juke or not. This simple scheme does not fully describe the complexity of a real arena game, but it is a good enough approximation to garner some very important insights about kicking and juking. Once we have created a game (in the game theory sense), we can construct a payoff matrix similar to the one we did above for rock-paper-scissors. Once we have the payoff matrix, we can find the Nash equilibrium of the game. So let's construct the payoff matrix for kicking and juking.

P1 Hold P1 Kick
P2 Cast \(P_\mathrm{HC}\) \(P_\mathrm{KC}\)
P2 Juke \(P_\mathrm{HJ}\) \(P_\mathrm{KJ}\)

You will notice that instead of discrete outcomes (win/draw/lose), a number of letters have taken their place. These letters denote unknown win probabilities. Let's unpack what we're trying to do here. Player 1 and Player 2 are opponents on different teams. Player 1 belongs to Team A, and Player 2 belongs to Team B. The above matrix describes what happens when Player 2 casts a spell, and Player 1 holds their kick. The goal of the payout matrix is to describe the effect that this combined set of actions had on the arena game. Following these two actions, the probability that Team A will win the game is \(P_\mathrm{HC}\). This might be 50% or 60% or some other value, but for now we just denote it with the letter \(P\) and the subscript HC (hold kick-cast). If, on the other hand, Player 1 lands a kick, there is a \(P_\mathrm{KC}\) chance that Team A will win the game (the subscript KC here denotes kick-cast). When Player 2 tries and fails to juke Player 1 (i.e. Player 1 does not kick), there is a \(P_\mathrm{HJ}\) chance that Team A will win. Finally, when Player 2 successfully jukes a kick, there is a \(P_\mathrm{KJ}\) chance that Team A will win.

It might seem difficult to populate this table with sensible values, but we know quite a few things about WoW PvP which will allow us to choose some values that make sense. In order to populate the payout matrix, we can consider specific types of scenarios and work through some sensible probabilities for that scenario.

Scenario 1: Optimal kick frequency when you're about to have your face melted

Let's first consider the case where our opponent is casting a decisive ability. This could be a well-timed CC that is likely to end the game in their favour, or a very high damage ability such as a Chaos Bolt or a Greater Pyroblast. The way that this manifests itself in the payout matrix is that \(P_\mathrm{HC}\) is very low. Indeed, let's consider a case where you have only a 5% chance of winning if the cast goes off (\(P_\mathrm{HC}\) = 5%). If you manage to kick the cast, you might still not be in perfect shape, but this does give you a chance to recover. Let's set \(P_\mathrm{KC}\) = 30%, i.e. you are still an underdog, but the game is winnable. What would we expect if we get juked? Well, our opponent did cancel their cast so it bought us a few more seconds, but we're still probably in pretty bad shape. Let's say that our chance of winning when we get juked is only 15% (\(P_\mathrm{KJ}\) = 15%). Finally, when our opponent tries to juke us and we don't kick, we're in better shape than if we got juked (since we still have our kick). On average, we are probably not in as great of a shape as if we had kicked the the kick. Therefore, the final value - \(P_\mathrm{HJ}\) - should be between 15% and 30%. Let's choose a value of 25% (for the sake of the argument). In general, this value will often be less than the win probability when you land a kick, and will rarely be significantly higher than it. Now we can populate the previous table with the values we've come up with.

P1 Hold P1 Kick
P2 Cast 5% 30%
P2 Juke 25% 15%

GTO kick frequency: 57.1%

Scenario 2: kicking a cast with moderate impact on the game

This is perhaps a more common situation. We're playing a DPS class in a relatively evenly matched game. On the other team there is a solid healer who's known to juke casts. We want to land as many kicks as possible, but we want to avoid getting juked. How do we ensure that we maintain our edge against the opponent team? Let's construct a payoff matrix. If we land a kick, we'll say that we have about a 60% chance of winning (\(P_\mathrm{KC}\) = 60%). If we get juked on our kick, we'll say that the state of the game returns to a coin flip, with us having a 50% chance of winning the game (\(P_\mathrm{KJ}\) = 50%). If we hold our kick and the healer gets a cast off, we don't lose all of our edge since we can still kick the next cast. Let's say in this case, there is a 53% chance of us winning (\(P_\mathrm{HC}\) = 53%). Finally, when we hold our kick and the healer jukes, this must be better than when the healer actually gets a cast off (but probably not as good as when we land a kick), so let's say it's around 57%.

P1 Hold P1 Kick
P2 Cast 53% 60%
P2 Juke 57% 50%

GTO kick frequency: 28.6%

A surprising and perhaps counter-intuitive result is that if the impact of a kick is large, you should actually kick less often. This is because the value of successfully juking a kick is now far higher, and your opponent is therefore incentivised to juke more often. Consequently, the optimal response to this is to kick less often. If, on the other hand, the impact of a kick is low, your opponent is incentivised to juke less often (due to there not being much value in a successful juke) and consequently, you should kick more often. Thus, slightly paradoxically, you should kick often for low impact kicks (e.g. 40% of casts), and less often for high impact kicks (e.g. 25% of casts). The figure below shows this for Scenario 2, but with varying kick impact.

As we move to from left to right, our kicks are becoming more impactful. Consequently, we want to kick less often to prevent our opponent from getting an advantage over us by constantly juking our kicks. I've also highlighted the regions with probable and improbable kick impacts (based on the other values specified in Scenario 2).

What about other scenarios?

There are many other scenarios and probabilities that you might want to consider. For that purpose I've created a GTO calculator which allows you to construct your own payoff matrix and get an optimal kick frequency out. Note that a fair bit of caution needs to be exercised when constructing a payoff matrix since not all payoff matrices will be valid. I would therefore recommend starting off with the baseline numbers provided and adjust each number accordingly.

GTO kick calculator

P1 Hold P1 Kick
P2 Cast
P2 Juke

GTO kick frequency:

Concluding remarks

Based on the probable region of various scenarios (including Scenario 2, above), it seems that a kick probability of around 30% to 40% provides a solid baseline strategy. Does that mean you should never kick more than 40% of the time? If you know that your opponent doesn't juke, you can still quite comfortably kick most if not all of their casts. At higher rating, however, it is unusual for players not to juke, and it is unusual for people to have egregious flaws in their juking strategy. In addition to the focus on game theoretic play in poker, there is also a large focus on identifying exploitable tendencies both in individuals and at the population level. If we know that a person jukes too much, we can exploit this by not kicking as much as we should if we were playing optimally from the perspective of game theory. Contrariwise, if we know that a person doesn't juke enough, we can kick more often than suggested by game theory. We can apply the same type of reasoning at the population level to tailor our strategy: if we believe people at our rating are in general juking too much, we might make an adjustment and kick less than suggested by game theory in order to exploit our opponent's suboptimal play. This type of strategy - a GTO base with exploitative adjustments - is the bread and butter of some of the best poker players on the planet.

The derivation for the Nash equilibrium for kicking is obtained by observing that the expectation for a Nash equilibrium strategy should be the same when the opponent jukes all the time and when the opponent jukes none of the time. Let \(Q_\mathrm{juke}\) denote the probability that the opponent will juke, and \(Q_\mathrm{kick}\) denote the probability that we will kick. We assume that these events are independent of one another. Using the notation from the article, we can now write the expectation as $$ \begin{align*} E &= Q_\mathrm{kick} Q_\mathrm{juke} P_\mathrm{K,J}\\ & + Q_\mathrm{kick} (1-Q_\mathrm{juke}) P_\mathrm{K,C}\\ & + (1-Q_\mathrm{kick}) Q_\mathrm{juke} P_\mathrm{H,J}\\ & + (1-Q_\mathrm{kick})(1-Q_\mathrm{juke}) P_\mathrm{H,C} \end{align*} $$ Note that \(P\) refers to game win probabilities if the event happens (i.e. the value of the outcome) and \(Q\) refers to event probabilities. \(Q_\mathrm{kick}Q_\mathrm{juke}\) therefore refers to the probability that we will kick and our opponent will juke.

In order to obtain the Nash equilibrium, we establish an equality between the expectation for \(Q_\mathrm{juke} = 0\) and \(Q_\mathrm{juke} = 1\). $$ \begin{align*} E_\mathrm{juke} = Q_\mathrm{kick} P_{K,C} + (1-Q_\mathrm{kick})P_{H,C}\\ E_\mathrm{cast} = Q_\mathrm{kick} P_{K,J} + (1-Q_\mathrm{kick})P_{H,J}\\ E_\mathrm{juke} = E_\mathrm{cast}\\ \end{align*} $$ Put the two equations together and solve for \(Q_\mathrm{kick}\) $$ \begin{align*} Q_\mathrm{kick} P_{K,C} + (1-Q_\mathrm{kick})P_{H,C} = Q_\mathrm{kick} P_{K,J} + (1-Q_\mathrm{kick})P_{H,J}\\ Q_\mathrm{kick} P_{K,C} + P_{H,C} - Q_\mathrm{kick}P_{H,C} = Q_\mathrm{kick} P_{K,J} + P_{H,J} - Q_\mathrm{kick}P_{H,J}\\ Q_\mathrm{kick}(P_{K,C} - P_{H,C}) + P_{H,C} = Q_\mathrm{kick}(P_{K,J} - P_{H,J}) + P_{H,J}\\ Q_\mathrm{kick}(P_{K,C} - P_{H,C} - P_{K,J} + P_{H,J}) = P_{H,J} - P_{H,C}\\ Q_\mathrm{kick} = \frac{P_{H,J} - P_{H,C}}{P_{K,C} - P_{H,C} - P_{K,J} + P_{H,J}} \end{align*} $$