Does complexity reduce coordination?

Abstract

Herein it is shown that increased complexity does not necessarily imply more coordination failure. Experienced people playing a 4-player spatial grid game with over 68 000 strategy choices and (68 000)⁴ potential outcomes were as likely to find the Pareto dominant Nash equilibrium as in a corresponding normal form game.

Acknowledgements

Thanks to the Institute for Environmental and Natural Resources at the University of Wyoming for the financial support. Comments from Amnon Rapoport, Peter Frykblom, Stephan Kroll, and participants at the workshop on ‘Bounded Rationality’ at CIRANO and the 10th annual Ulvön workshop have been helpful.

Notes

Other non-normal form games include Rapoport and Erev's (1998) market entry games, Metha et al.'s (1994) circle-to-square spatial assignment games, and Lim et al.'s (1994) overlapping generations game.

See for instance Cooper et al. (1990), Van Hyuck et al. (1993), Crawford (1998), and Camerer (2003).

All experimental instructions are available from the authors on request.

When a cell is browned out the subject gives up the market value of the cell. Now he earns the various benefits, the total value depending on the spatial configuration of own and other players’ brown cells.

Possible strategy choices are calculated using a combination – a selection of items in which order does not matter. The formula for a combination is N!/(X!(N-X)!), where N is the total number in the set and X is the items that one can select. Here N is the number of parcels, 25, and X is the number of parcels a player must brown out, X was an element of the set [0, 5]. The equation is (25!/5!20!) + (25!/4!21!) + (25!/3!22!) + (25!/2!23!) + (25!/1!24!) + (25!/0!25!) = 68 406.

The only restriction on communications was to prohibit threats or vulgarity. Cheap talk could say the subject's strategy choice or could be unrelated to the experiment (e.g. how's the weather?).

The dominant Nash equilibrium requires subjects to brown out cells with market value of $10. In rounds 1–10 in session 1, 51 messages were sent – 19 said to play the payoff dominant strategy, 8 suggested inferior strategies, and 24 had context unrelated to the game. In rounds 11–12, all 11 messages had the payoff dominant strategy. In rounds 13–20, 2 messages sent the dominant strategy, and 23 of 25 messages were game-unrelated. In session 2, for rounds 1–10, 21 of 47 messages said the payoff dominant strategy, 23 suggested other strategies, and 3 were game-unrelated. In rounds 11–15, 18 of 19 messages had the payoff dominant strategy; for rounds 16–20, 2 messages had the dominant strategy, and 16 of 18 messages were game-unrelated.

In the normal form game, subject's communicated their strategy choice by indicating the action, 1, 2, … , 8, they planned to play, which likely resulted in less confusion in the interpretation of messages. In rounds 1–10, 73 of 74 messages sent in session 1 and 63 of 70 messages sent in session 2 indicated the payoff dominant strategy (4). In rounds 11–20, all messages sent indicated the payoff dominant strategy, 73 in session 1 and 75 in session 2.

The chi-square probability test assumes samples are randomly drawn from one population or two separate populations. We use the chi-square probability test for all hypotheses tests between treatments.

The chi-square test for independence between rounds establishes independence for rounds 3–20 in the normal form game treatment, and rounds 11–20 in the grid game treatment. In the grid game treatment, all subjects converged to the dominant Nash equilibrium as a group, and the payoff dominant strategy individually by round 10. For the normal form game, 6 of 8 pairs converged to the dominant Nash equilibrium by round 11 and 7 of 8 by round 16.

One player in the normal form game indicated through his cheap talk communication he would play the payoff dominant strategy but then played a different strategy. An analysis without the idiosyncratic observations of this person does not change the results at both the group and individual levels. Tables 2A and 2B report these test results.