## 1. Introduction

Human performance in complex tasks is often a combination of internal planning and responding appropriately to the environment. Nevertheless, cognitive models of complex tasks typically focus on the mental planning aspects, and fail to take into account that the external world can heavily influence the control of behavior.

The role of the environment was first recognized in robotics (Brooks, 1991), but it was later extended to human cognition to form *embodied cognition* (e.g., Clark, 1997). However, in more complex tasks, it is clear that the control of behavior is not entirely in the environment. The challenge is, therefore, to understand how control is shared between goal-driven planning and processes that are driven by perceptual input. Moreover, the balance between goal and perceptually driven control is likely to change with expertise (Kirsh & Maglio, 1994). The approach we take in this article follows the *threaded cognition* theory of multitasking (Salvucci & Taatgen, 2008). We will assume two parallel processes: a bottom-up visual process that scans the visual field on the basis of saliency and similarity, and a top-down planning process that tries to achieve the goal but also biases the bottom-up process. The interaction between the two processes follows the central idea in threaded cognition that there is no overall executive process that balances parallel goals. Instead, the two processes alternate in using the cognitive resources (e.g., vision, declarative memory [DM], procedural memory, etc.). Changes in the balance between the two occur if one process benefits more from learning than the other and therefore makes more efficient use of the resources available to it.

Finding an appropriate task to study the cognitive aspects of human behavior in real-life situations is not easy. However, games provide environments that often require the same type of complex processes that are usually involved in real-world situations (Green & Bavelier, 2004). This has the advantage that the behavior of a player can be studied in a controlled environment. These qualities make games on a computer an ideal tool for studying complex cognitive processes. One such game is the card game SET.^{1}

The SET card deck consists of 81 cards. Each card differs from other cards by a unique combination of four attributes: color, number, shape, and shading. Each attribute can have one of three distinct values: red, green, and blue for the color; open, solid, and textured for the shading; one, two, and three for the number; oval, rectangle, and wiggle for the shape. The gameplay for SET is relatively simple. At any moment in the game, 12 cards are dealt face up, as is shown in Fig. 1. From those 12 cards, players should find any combination of three cards, further referred to as a set, satisfying a rule stating that in the three cards the values for each particular attribute should be all the same or all different. We will further refer to the number of attributes for which the three cards in the set have different values as the set *level*. A level 1 set has only one attribute with three different values, but three attributes with identical values. Correspondingly, there can be sets of level 2, 3, or 4. Fig. 1 shows an example of a level 1 (different shape) and a level 4 set (all attributes are different). In a similar manner, we can quantify *perceptual similarity* of two cards as the number of attributes that are shared between the two. For example, the cards in a level 1 set have a perceptual similarity of three among each other as they have three attributes with identical values. Cards in a level 4 set have a perceptual similarity of 0 because all attribute values are different.

In the regular game, when a set is found, the corresponding set cards are picked up and replaced with new cards from a deck. After the deck runs out, the player with the most cards wins. Even though a regular game of set consists of multiple rounds, we will refer to a “game of set” in what is normally a single round: finding a set in 12 displayed cards.

There are several advantages of choosing SET as a target game of study. First, SET has an appealing simplicity of the game dynamics. The game has very simple rules to follow and a relatively static game environment. Despite the simplicity, SET requires complex cognitive processes, including pattern recognition, visual processing, and decision making. Previous studies on SET have established that both cognitive and perceptual processes are important (Jacob & Hochstein, 2008; Taatgen, Oploo, Braaksma, & Niemantsverdriet, 2003). Without consideration of both of them in combination, important information in understanding of how players play the game will be inevitably lost. As such, the game of SET provides an excellent opportunity to study the dynamics of such processes in a relatively simple environment.

Next, the game is quite unpredictable in its structure, and players are not likely to replay the exact same sequence again. There are 7*10^{13} possible combinations of 12 cards, which makes it highly unlikely that players will play through the same 12 cards again. There are also 1,080 different sets. This means that even experienced players will periodically have to find a set they have never encountered before.

Finally, game difficulty can differ significantly based on a player’s strategy. Given an array of 12 cards with a single set in it, a player may choose to compare every possible combination of three cards. There are 220 possible combinations, and a probability of finding a set with random choice is 1/220. However, a player may also consider combinations of two cards as a pair uniquely defines a third card. In that strategy, a player would pick two cards, would then determine what the third card should be to complete a set, and would then see whether the predicted third card is actually among the remaining 10 cards. There are 66 possible pairs, and the same set is defined by three different pairs. Therefore, a probability of finding a set with a random choice of a pair is 1/22. However, with an optimal search strategy, a player still has to consider a maximum of 54 pairs before finding a set. This is a significant decrease in complexity compared with a strategy where a player has to compare every combination of three cards.

The above two strategies are both top-down in the sense that they do not take into account what the properties of the particular array of 12 cards are. However, players are likely to be using perceptual processes and clues, such as visual grouping and visual similarity, to decrease complexity or speed up the search. As an example, suppose that there are eight red cards and two cards each for blue and green. Furthermore, let us assume that a player is using similarity in color to find a set. Blue and green cards cannot have a set as there are only two cards in each group. There are 56 combinations of three cards among red cards and 32 combinations of three cards with different colors. It is already a significant decrease in complexity from 220 to 88 possible combinations and a 2.5 time increase in a chance probability of finding a set. Chance probability of finding a set among red cards is even higher 56/88 or about 2/3. This leverage in a chance probability only comes from a larger group size for red cards. For example, if there is an even split of four cards for each color, then the chance probability of a set being among cards of the same color is only 4/76. As will be discussed next, players actually exploit the advantage of a larger group size.

There are two studies directly relevant to the work in this article. Jacob and Hochstein (2008) did several experiments with human subjects playing SET on a computer without any opponent. Each experiment was designed to test a particular aspect of the game including a strategy of playing the game, dependency of the performance on the set level, attribute preference, and the learning. Taatgen et al. (2003) also did similar experiments aimed at studying the strategy of playing the game and developed a computer model of a human player.

Jacob and Hochstein (2008) demonstrated that SET players prefer to look at perceptually similar cards, and, for the comparison of the cards, mainly rely on the perceptual processes such as similarity-detecting process. According to the authors, bias to the perceptual similarity and corresponding bottom-up processes can explain why players need less time to find lower level sets than higher level sets. Taatgen et al. (2003) also reached the conclusion that the perceptual elements play a greater role in finding lower level sets. They suggested a strategy where a player looks at an arbitrary first card then at a second card that shares an attribute value. Next, the player predicts the third card and determines whether that card is one of the remaining 10 cards. Taatgen et al. (2003) also hypothesized that the choice of the first card might not be arbitrary in some cases. They proposed that players try to find the set among the cards that have an attribute value occurring in more than half of 12 cards. For example, if there are many red cards, it is attractive to search for a set among those cards. Taatgen et al. (2003) implemented this strategy in an Adaptive Control of Thought–Rational (ACT-R) model. However, the data they collected did not have enough detail to determine whether subjects used such a strategy.

Jacob and Hochstein (2008) proposed a generalization of the above strategy based on the notions of the most abundant value and the most abundant-value group. The former refers to an attribute value that occurs most, and the latter refers to the group of cards that have the most abundant value. They found that the sets belonging to the most abundant-value group are preferred to the sets outside of that group. In addition, the time required to find the set in the most abundant-value group decreased as the size of the group increased. Most abundant-value group was preferred to any other value group independently of the attribute type. Jacob and Hochstein (2008) suggested a *dimension-reduction strategy* where players try to reduce the four-dimensional search space to three by choosing to look at cards that have one or more attribute values in common. It was assumed that dimension-reduction strategy is primarily used with the most abundant value.