That is, 3 cards of one type when trying to count several different types of cards at once. This limit, which is observed even in subjects after much practice in the game, is contrary to the folklore of Bridge, Blackjack and other card games where people can presumably count much higher. The difference between Bridge and TRACS is that there are multiple (identical) copies of each card in the TRACS deck. So, while a Bridge player can count higher than 3 spades (or clubs or hearts or diamonds), the spades are all different (Ace, King, Queen, etc.). In other words, Bridge players have a richer context in which to count cards. In the case of Blackjack (Mezrich, 2002), the card-counting strategy is simply to count how many more low cards (2, 3, 4, 5, 6) have been seen than high cards (10, J, Q, K, A) while ignoring the mid cards (7, 8, 9). In other words, Blackjack players are essentially counting only one thing at a time.

The challenge in TRACS is to count multiples copies of more than one type at once, and this is where we found the mental limit of about 3 (plus or minus 1) for each type. Although 7 (plus or minus 2) is the famous capacity limit (Miller, 1956) for discriminating "unidimensional stimuli" (i.e., pitches in hearing), recent research (Cowan, 2001; Dehaene, 1997) suggests that a limit closer to 3 (plus or minus 1) exists for other tasks like "subitizing" (i.e., dots in vision). From the perspective of Natural Computation (Richards, 1988), mental limits like this are not just arbitrary constraints but rather reflect the natural structure of the world in which mankind evolved. In fact, there are computational arguments that suggest a "limitation" of 3 in one context (like seeing dots or counting cards) may actually be an "optimization" in other contexts (like hierarchical memory organization; see MacGregor, 1987).

The number 3 is a natural limit because it is the minimum number needed to represent a variety of natural structures. For example: (1) the dimensions of space: height, length, width; (2) the progression of time: start, middle, end; (3) the estimation of stuff: high, medium, low. In art, haiku poems have three lines (start, middle, end). For 3 other examples, consider 3 acts in the theatre, 3 books in a trilogy and 3 frames of a comic strip. In science, many classification schemes are based on threesomes like (1) solid, liquid, vapor; (2) earth, wind, fire; (3) animal, vegetable, mineral. Three other examples are arithmetic (x+y=z); physics (F=ma); psychology (id, ego, super-ego).

Computer Model

Our research on TRACS (Laboratory Tool) has generated data on how people make probabilistic judgments in a dynamic context. These data show a behavioral pattern that is consistent with the previously identified heuristic of "anchoring and adjustment" (Kahneman, Slovic & Tversy, 1982), but we make two new contributions. First, we measure human performance in a continuously "dynamic" task (card game), as opposed to the primarily "static" tasks (questions) studied in previous research. Second, we implement our models in computer simulations that provide more powerful explanations and predictions of anchoring and adjustment behavior.

Our computational model uses "fuzzy functions" and simple summing to simulate the cognitive processes of memory activation and evidence accumulation. The model (gray line in figure below) provides a good match to data (black line in figure below) on judgments versus time, especially compared to a simulated agent (dotted line in figure below) that counts cards and computes odds perfectly (i.e., with no mental limits). The details are documented in 1, 2, 3, More: An Accumulator Algorithm for Anchoring and Adjustment (Burns, 2003).

References

Burns, K. (2002a). On Straight TRACS: A Baseline Bias from Mental Models. Proceedings of the 24th Annual Conference of the Cognitive Science Society, 154-159 (Erlbaum). 

Burns, K. (2002b). Dealing with TRACS: The Game of Confidence and Consequence. Proceedings of the American Association for Artificial Intelligence, Symposium on Chance Discovery. 

Burns, K. (2003). 1, 2, 3, More: An Accumulator Algorithm for Anchoring and Adjustment. Proceedings of the 6th International Conference on Computational Intelligence and Natural Computing.

Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24, 87-185.

Dehaene, S. (1992). The Number Sense: How the Mind Creates Mathematics (Oxford University Press).

Kahneman, D., Slovic, P. & Tversky, A. (1982). Judgment Under Uncertainty: Heuristics and Biases (Cambridge University Press).

MacGregor, J. N. (1987). Short-term memory capacity: Limitation or optimization? Psychological Review, 94(1), 107-108.

Mezrich, B. (2002). Bringing Down the House: The Inside Story of Six MIT Students Who Took Vegas for Millions (Free Press). 

Miller, G. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63, 81-97. 

Richards, W. (1988). Natural Computation (MIT Press).

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