How Does Luck Influence the Board Game Sequence?

Losing in Board Games is rarely fun. And certainly not when luck decides the game. But how much influence does luck actually have in popular board games like Sequence? I played the game and looked at the numbers. This analysis can not be generalized to other players, not to mention Sequence variants with more than 2 players, as it is based only on matches between me and my partner.

What Is the Game About?

If you’ve never played Sequence, here’s a quick introduction:  

The game board is a 10 by 10 grid, where each space corresponds to a card (Figure 1). The game is played with two decks with a total of 104 cards. Each player is dealt a number of cards and takes turns playing one card and placing a token on the corresponding space on the board, then draws a new card from the deck. There are also some lucky cards:  

• Two-eyed Jack: Allows you to place a token anywhere. 4 cards in total. 
• One-eyed Jack: Removes one of your opponent’s tokens. 4 cards in total.

In this analysis, we focus on a two-player Sequence, where a player wins by being the first to make two sequences of five tokens in a row — horizontally, vertically, or diagonally.

Let us play

In the name of Statistics, I played 51 games with my partner. Going forward, we will refer to ourselves as Player 1 and Player 2. We recorded several variables, including the number of rounds, the winner, and the number of one-eyed and two-eyed jacks drawn in each game, excluding those drawn in the last two rounds unless they were played.

Distribution of jacks

Our first task on this meaningless journey was to answer the question: Did we make some lucky draws?

To that end, let N, X_1eye and X_2eye be the random variables which describe the number of rounds, the one-eyed cards and two-eyed cards seen by one player. If we assume a game lasts N = n rounds, then we can ask what is the expected number of one- or two-eyed jacks in such a game. Assuming the deck of 104 cards is randomly shuffled, and each player is dealt cards interchangeably, this question is similar to asking what is the expected number of successes in N draws without replacement, given a population of size 104. This is the well-known hypergeometric distribution, and since there are 4 one-eyed and 4 two-eyed jacks, it follows:

Using the same argument, we find:

where X  is the number of jacks (any type) seen by a player. The variance of the distribution is:

We can look at Figure 2 and verify that jacks drawn follow the distribution nicely with only 6.25% of extreme cases (i.e. outside the 95% prediction interval), which also reassures us that the cards are properly shuffled.

Our recordings provide estimates for the mean rounds E[N]≈19 and variance V[N]≈24 and using the tower property of conditional expectation we find:

and from the law of total variance, we find:

Empirically, we found:

for Player 1, and:

for Player 2, suggesting that we might have been more than average lucky with drawing Jacks.

What is the real value of jacks?

One thing is getting lucky cards, another thing is how much a lucky card (a jack) actually impacts the game. To investigate, we performed a simple logistic regression, with the aim of measuring the effect of one-eyed and two-eyed jacks on the winning chance, denoted p:

where ⍺ is the effect on winning from drawing a one-eyed jack and β is the effect on winning from drawing a two-eyed jack. The parameter ɣ is other contributions. We fit the model in R using the function glm. The estimates [p-value] are:

The only significant effect is from the two-eyed jacks, which also has the largest effect on the winning outcome. Specifically, we find that the odds of winning with a two-eyed jack is:

=2.27 times those with a one-eyed jack.

Metric of luck

Based on our regression, the value of a one-eyed jack and two-eyed jack, written v₁  and v₂ respectively, was determined by the equations:

For each player and each game i, we defined a luck score ( H_i ) as the total value of the player’s jacks. A score of H_i = 0 meant no luck (no jacks), while H_i = 1 meant maximum luck (all 8 jacks; 4 × v₁ + 4 × v₂).

We could further calculate the difference in luck ∆H_i = H_i^{player 1} – H_i^{player 2} between players. A score of ∆H_i = 0 meant an equal amount of luck (measured by the total value of the jacks), while ∆H_i = 1  meant that Player 1 drew all 8 jacks. When ∆H_i = -1 it meant that Player 2 drew all 8 jacks. We agreed on some (semi) arbitrary bounds:

• ∆H_i = -0.5 corresponds to very lucky (player 2)
• ∆H_i = -0.25 corresponds to lucky (player 2)
• ∆H_i = 0 corresponds to no luck (either player)
• ∆H_i = 0.25 corresponds to lucky (player 1)
• ∆H_i = 0.5 corresponds to very lucky (player 1)

We found that when a player experienced luck, the chance of winning was dramatically increased. Only in 7 out of 51 games did the unlucky player manage to win. See Figure 3, where the compelling image is shown.

Metric of skill

Now, it would be controversial to claim that two-player Sequence is no different from a coin toss. So how much is skill? Here we defined a skill score E_i  for each game i using the following formula:

The expression 1 + H_i represents the number of skill points we want to award the player. However, we will only award points if the player has not been lucky. This is described by the indicator function ↿_{{∆H≤ 0}} for Player 1, and ↿_{∆H≥0} for Player 2. Also, we will only add skill points if the player actually wins, described by the indicator function ↿_{{Winner = Player 1}} for Player 1 and ↿_{{Winner = Player 2}} for Player 2. Thus, assuming a player wins, the skill score is zero, if the player was lucky, 1 if either player was equally lucky, and 1 + H_i otherwise. The skill score is also zero, if the player loses.

Our best estimate of a player’s skill in Sequence is the average skill score Ē over all games where E_i ≥ 1  (i.e. where the player has been skillful). For both Player 1 and Player 2, there are m = 7 games where the players have been skillful, as seen from Figure 3. The score is then:

In Figure 4, we used a technique called bootstrap resampling to create 5 million replicas of the average skill score Ē for both Player 2 and Player 1, based on our games. We then computed 5 million differences ∆Ē =Ē_{player 1} – Ē_{player 2} and plotted the histogram, along with a 95% (bootstrap) confidence interval, and the mean difference.

The figure shows significant uncertainty about which player performs the best. In particular, it is noted that the confidence interval contains 0, which means that at a 5% significance level, we cannot reject the hypothesis that the two players are equally skilled.

A game of chance?

We have demonstrated that luck plays a major role in our games of Sequence. When either me or my partner drew enough jacks, victory was nearly certain. To evaluate the ability to strategically navigate a fair game or one with opposition, we developed a skill score. The analysis of 51 games showed no significant difference in skill between my partner and me. Other luck factors, such as card order, are not accounted for in our model, meaning that the games used to calculate the skill score may themselves be influenced by elements of luck. If this is the case, the game is more affected by chance than our analysis suggests.

Our study is based on 51 games, which is a relative low number. These games were played by me and my partner and cannot be generalized. Other players with greater or lesser strategic sense will likely affect the outcome differently. Still, our analysis revealed that two-eyed jacks displayed a significant and substantial effect on the chance of winning, suggesting that they play an important role, regardless of the players. Although one-eyed jacks were not statistically significant, we still considered them to be lucky cards, as they are logically expected to have some influence. Ultimately, while our findings suggest that the strongest predictor of victory is the number of two-eyed jacks, further analysis over many more games with a broader range of players would obviously be needed to fully understand the dynamics at play. For example, it might be hypothesized that equally skilled players profit significantly more from luck, as opposed to players of uneven skill.