ABSTRACT

The literature on incrementalism is concerned with decentralized adaptation by decisionmakers with limited information to the complexity of the decision environment. However, thespecific aspects of limited information and complexity are often left unexamined, leavingconclusions ambiguous. In order to understand better the effect of limited information andcomplexity on decentralized decision making, we design an experimental game in whichsubjects play in the basic Cournot duopoly setting. This setting is similar to the incrementaldecision example given by Lindblom (1965). The game provides the players with no infor-mation about what each player’s decisions are other than the history of their mutual inter-action, thus guaranteeing imperfect information. We also manipulate two factors of com-plexity: the risk of making a bad decision and the strategic uncertainty of a particular courseof action. Through these manipulations, we seek to induce and better understand incre-mentally adaptive behavior by the participants.

INTRODUCTION

There has been much literature written on decision making in decentralized environmentswith limited information. Decision makers are limited in both what they can control in thedecision domain and what information they can obtain. Under these circumstances, decisionmakers adapt to decisions made by others in a decision process, not in a singular event. Theoverall outcome of the decision process is the collective result of the interactions among thedecision makers. The process consists of sequences of many, often relatively small movesmade by all players. Decision makers in this environment are said to “incrementally adapt.”

Adaptive Incrementalism and Complexity:Experiments with Two-Person CooperativeSignaling GamesJack H. KnottUniversity of Illinois at Urbana-Champaign

Gary J. MillerWashington University

Jay VerkuilenUniversity of Illinois at Urbana-Champaign

DOI: 10.1093/jopart/mug023

Journal of Public Administration Research and Theory, Vol. 13, no. 3, pp. 341–366© 2003 Journal of Public Administration Research and Theory, Inc.

We wish to thank Janet Glaser, Vice Chancellor for Research, and the Research Board for generous funding as well as help navigating through the paperwork necessary for research involving human subjects, James Kuklinski andHamish Gow for important advice at various points, Mary Parker, the students and instructors of the classes and athletic team that made up the subject pool, Jennifer Jerit and Phil Habel for help administering the experiment, andthe participants on panels at the American Political Science Association conference in August 2000 and the MidwestPolitical Science Association conference in April 2001, where previous versions of this article were given.

The study of incremental adaptation spans several disciplines. Scholars of public bud-geting explain the incremental allocations of budget resources from one year to the nextusing a model of interaction among the chief executive, public agencies, and the legislature(Wildavsky 1964; Crecine 1969; Davis, Dempster, and Wildavsky 1966). Similarly, publicpolicy and bureaucratic-politics scholars have examined the “muddling through” and “dis-jointed incrementalism” of policy making among interest groups, executive agencies, andthe legislature (Lindblom 1959; Bendor and Moe 1985; Rourke 1984). In most of thesemodels and in organization theory, the literature has focused on “bounded rationality” andtrial-and-error decision making among organizations and the environment (Simon 1957;March and Olson 1976). Jones (1999) has an excellent review.

In each case, incremental adaptation is explained by the complexity of the problemscombined with the cognitive limitations, or bounded rationality, of decision makers (Simon1957; Williamson 1985). In Crecine’s (1969, 38) simulation of the budget process, for ex-ample, the entire budget process is viewed “as an organized means for the decision-makerto deal with the potential complexity of the budgetary problem.” In contrast to the idealizedmodel of homo economicus, boundedly rational individuals acquire information throughtrial-and-error processes, basing their decisions on past information, standard operating pro-cedures, and projections from the current state of affairs. Often they “satisfice,” that is,search for proximate, locally optimal, or even tolerably satisfactory solutions near the cur-rent alternatives. They constrain the decision space by “pruning” the decision tree of alter-natives and often cannot form transitive preference orderings of alternatives, even whenconsidering all of them (Simon 1957, chapter 14). Steinbrunner (1974), in comparing in-crementalism to a thermostat control, indicates that goals become aspiration levels that riseor fall depending on the success or failure of past efforts, and there is no attempt rationallyto integrate different values. March and Olson’s (1976) “garbage can” models of institu-tional choice elaborate this notion.1

Despite its wide use as a concept across disciplines, the incrementalism literature lacksa firm connection with the literature in institutional economics. A major work such asWilliamson (1985), however, demonstrates the considerable overlap between the two liter-atures, and, indeed, in his discussion of contracting he frequently cites Simon’s (1957)bounded rationality. Although both literatures deal with very similar problems, they do sofrom different perspectives. The incrementalism literature in organization theory and pub-lic administration has emphasized the cognitive limitations of decision makers to cope withthe ill-defined nature or enormity of problems. Presumably, with faster computers, better-educated and smarter decision makers, and better models, decision making would becomeless incremental. Even the quantitative political science literature on budgeting (e.g., Davis,Dempster, and Wildavsky 1966) focuses on the enormousness of the budget and on meas-uring the size of the increment, not on the politics of interaction among the actors. Thispoint was raised by McCubbins and Schwartz (1984) when they considered two differentmonitoring styles for members of Congress, one of which looked incremental but was moreconsistent with their incentives, that is, chosen rationally.

In contrast, institutional economics emphasizes the strategic aspects of such situationsmore than cognitive limits. Coase (1937) provided the fundamental insight, namely, thattransaction costs are the essential component for understanding the outcomes of joint deci-sions. An important element of transaction costs is the bargaining and strategy for producing

342 Journal of Public Administration Research and Theory

1 For an excellent review of critiques of incrementalism, see Lustick (1981) and Bendor (1995).

coordinated outcomes. Similarly, Tsebelis (1989) shows how the actions of one decisionmaker alter the choices of the other decision maker through mutual interdependence, intro-ducing probabilistic uncertainty through mixed strategies. This strategic aspect is found inearly writings on incrementalism, such as Lindblom (1965) and Wildavsky (1964), but isdiscussed at a general level with examples and anecdotes. Careful definitions of strategic in-teraction are not provided at a level of detail sufficient to make testable statements. Empiri-cal studies of incrementalism have tended to focus on measuring whether a decision processis incremental without showing why and how it occurs.

In this article we are concerned with how strategic interdependence explains incre-mental adaptation. We address this question by designing and implementing an experi-mental game that adds levels of risk and uncertainty to the strategic interaction of a de-centralized decision situation. We are interested in determining how these aspects ofcomplexity in the decision situation affect incremental adaptation in decision making. Theremainder of the article is divided into six sections: (1) a rationale for the game we havechosen to examine, (2) details of the experimental design for testing the game’s predictions,(3) the laboratory setting of the experiment, (4) the results of the experiment, (5) a discus-sion, and (6) the conclusion.

Incremental Adaptation in a Duopoly Game

The Duopoly GameTo study the effect of strategic interdependence on incrementalism requires a decisionsetting precise enough to make testable predictions. In The Intelligence of DemocracyLindblom (1965, 33) describes a two-person game for explaining the requirements for adap-tive incremental adjustment. A decision maker, let us call him Xavier, simply adapts to thedecisions of the other decision maker, whom we will name Yolanda, without directly nego-tiating or threatening a desired outcome. In effect, each decision maker decides in his or herown dimension and ignores the other dimension entirely, adjusts to the other dimension inorder to improve his or her own outcome, or adjusts to his or her own advantage but showssome concern for the effects on the other decision maker. Lindblom limits the degree ofdiscussion, negotiation, and bargaining between the two players in this most basic incre-mental adaptation game.

This kind of two-person game can be thought of as an institutional structure in whichone person is allowed to make nonamendable proposals in one dimension and another playeris allowed to make nonamendable proposals in another dimension. Suppose, for example,that the institutional rules let player Xavier choose an action from the set X, and playerYolanda has been handed the authority to choose an action from set Y. This can be thoughtof as a two-person game, but it can also be thought of as a generalized institutional structure.As Shepsle (1979) says, institutional rules allocate jurisdictional and amendment-controllingauthority to various actors.

In this article, therefore, we are concerned with a two-person signaling game, which iscomparable to a duopoly situation in economics. We examine this model because the settingthat Lindblom (1965) describes seems to be that of two-person game theory and appears tobe quite comparable to how duopolies adapt. In a duopoly, two firms that jointly serve amarket must determine their outputs. Given a particular level of overall market demand,each firm decides how much to produce based on what it believes its competitor will pro-duce, so the market clears.

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 343

Although a duopoly game emphasizes strategic interaction, we want this interaction tooccur in the context of a complex problem. In Wildavsky’s (1964) book The Politics of theBudgetary Process, for example, the author includes sections on calculation and strategy,emphasizing both the cognitive difficulty of making choices over the magnitude of the fed-eral budget as well as the political strategy of decentralized interaction. To this end, weembed the duopoly game in a context that allows for several choice options for each play.We also build in a sequence of choices over time to establish a decision process, not just asingular choice. This multiple set of choices and the sequential decision process are heldconstant across games.

We then focus our attention on the risk and uncertainty of the strategic interdependencebetween the two players, which occurs even when the players understand the game com-pletely. Situations of strategic interdependence involve varying degrees of risk and uncer-tainty. Risk is incurred when one’s rival in the game chooses an option that is a gain for him-self or herself, but when the choice is combined with one’s own choice, the mutual resultproduces a loss for oneself, or vice versa. To add to the risk, it is possible to increase the sizeand alter the probability of loss. Uncertainty occurs when players have more than onereasonable strategy to play the game. To add to uncertainty, we can alter the structure of thepayoff scheme in the game. With more than one plausible strategy, coordination of movesbecomes more difficult, and players may not as readily understand their opponent’s set ofreactions.2

Decision BehaviorWhat kinds of decision behavior do we expect to find in the duopoly game? The basic gamesituation is one in which each player has private information not shared by the other player.Because a player does not know everything, it is necessary for the player to form beliefsabout this private information in order to make choices. To try to predict the different formsof decision behavior in the duopoly game, therefore, we need to understand the beliefs thatplayers hold about the probability of different moves by the other players under differentconditions of uncertainty and risk. When it is feasible we also want to formalize a player’sbelief structure in order to better identify the player’s strategic choices.

Cournot AdjustmentWe hypothesize that, in situations in which strategic interaction entails relatively low riskand low uncertainty, simple maximizing rational choice algorithms will characterize adaptivebehavior. Under these conditions, players can hold fairly confident beliefs about the proba-bilities of their rival’s intentions. The process of converging to the equilibrium point, thoughrequiring interdependent choices, presents no major uncertainties or risks. Some of the bet-ter strategic players will reason forward through the set of moves to the equilibrium point and

344 Journal of Public Administration Research and Theory

2 Some may object to our definition of uncertainty, which differs somewhat from the one commonly used by gametheorists to denote a game in which there is a move by “nature.” To some extent our games do have this elementexperimentally because the level of risk-acceptance and the cognitive ability of a player’s partner is random.However, this is not under our control, so it is not something we can manipulate. Nevertheless, we believe, followingTsebelis (1989), that there are two sources of probability in a game: unpredictable moves by “nature” and the strategicinteraction of the players themselves. We increase uncertainty in the strategic sense by introducing more than oneattractive strategy and are thus using the term uncertainty in a broader way. It is also important to recall that thegames are of limited information.

either choose that point right away or follow a few steps to get there. We thus expect to findquick updating of information on the other player’s moves, confident assumptions about theother player’s likely next move, and a maximizing path to the equilibrium point.

The classic solution to the duopoly problem, which adopts this maximizing path, isassociated with Augustine Cournot, a French economist who in 1837 addressed the partic-ular two-person interactive game that arises in a duopoly. Fouraker and Siegel (1963), whoconducted a set of experiments to test bargaining behavior in duopoly situations, found thatunder conditions of incomplete information the results supported the Cournot solution. Thismodel thus has empirical support, matches the Lindblom description, is found in every eco-nomics textbook that treats the duopoly problem (e.g., Henderson and Quandt 1980 orPindyck and Rubinfeld 2001), and is considered a standard solution for this kind of situation.Furthermore, the Cournot solution can easily be extended to predict a path of adjustment,which gives us a hypothetical baseline path against which to compare observed behavior.

Cournot assumed that each player would select the outcome that maximizes his ownpayoff given any known choice for his opponent. He labeled this set of choices a “reactionfunction.” The reaction function notion requires that a player know what the opponent’schoice will be. The assumption that a Cournot decision maker adopts is that “each duopo-list acts as if his rival’s output were fixed” (Henderson and Quandt 1980, 215). That is, eachperson assumes that his opponent will pick in the upcoming period exactly what he pickedin the previous period. The assumption of the Cournot adjustment model, namely that one’sopponent will make the same move in the next play as in the last play, is a rational assump-tion when no other information is available. The assumption allows for a maximizing strat-egy under constraint. The Cournot solution is, therefore, a game in which all probability isassigned to the statement “my competitor will play the same as last time.”

The Cournot model, however, ignores the risks and uncertainties of interdependence.In the first play of the game, the Cournot player operates under uncertainty because theother player has not yet made any choice. The player in the first move will thus have littlereason to assume anything other than a random opponent. But in the second play of thegame, the Cournot player suddenly and completely switches from decision making underuncertainty to game-theoretic strategic decision making against a rational, self-interested,and therefore predictable, opponent. Under conditions of risk and uncertainty, this shift maynot be reasonable.

Sequential Bayesian UpdatingBayesian updating is an alternative general framework in which to understand choices in de-centralized games. A player’s beliefs are represented by a subjective probability calculus,that is, by Bayesian probability (Dixit and Skeath 1999; Gintis 2000). The Bayesian playeris less confident that the immediate prior move of the other player is 100 percent predictiveof the next move. Having no prior knowledge of the other player’s preferences at the startof the game, the Bayesian player begins with a uniform prior. It is assumed that she willchoose the first move rationally based on a maximum expected value (MEV) criterion andhence will choose the MEV row. In the second play of the game, the Bayesian player willmove based on the average of the prior probability distribution and the probability distri-bution of the first observation. For all subsequent plays of the game, the Bayesian playerwill update beliefs about probabilities based on all prior observations using a weighted av-erage with recent moves weighted more than earlier moves. Over time the Bayesian players

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 345

converge to the Nash equilibrium. As the mathematics is rather complex, the calculations forthese Bayesian moves are provided in appendix 2.3

Incremental AdaptationCournot and sequential Bayesian convergence are examples of change over time and mutualadjusting behavior. Are they then examples of incrementalism? Observations of Cournotbehavior in real-world decision settings may look like incrementalism because of the mu-tually adjusting behavior over time. Similarly, a sequential Bayesian player could choose in-crementally, depending on the structure of the game. Following Bendor and Moe (1985),however, we want to include in the definition of incremental adaptive behavior more limitedmoves in the direction of greater utility. Thus adaptive incrementalism includes nonopti-mizing behavior based on limited information, whereas Cournot behavior is based on max-imizing behavior and confident predictions about the opponent’s behavior. For this reason,we are led to try to understand what nonoptimizing, incremental adaptive behavior wouldlook like and what conditions would cause it to occur.

We hypothesize that under conditions of greater uncertainty and risk, players will movein smaller steps because of the greater difficulty of establishing credibility and commitment(Dixit and Skeath 1999, 302–13). They will attempt to take small steps to establish a patternof interaction and be rather cautious about updating information about the decision patternof the other players. Hence they will not move away from their initial choices very quickly,and when they do move away, they will move in a nonoptimizing adaptive fashion, withsmaller steps than might be predicted by rationality assumptions. According to Schelling(1960, 85), players attempt to create stable expectations about the other player’s motives andintentions in order to establish mutual trust that the rival will not double-cross or renege oncooperative moves. He writes, “Players have to understand each other, to discover patternsof individual behavior that make each player’s actions predictable to the other; they have totest each other for a shared sense of pattern or regularity.” This stability of “convergent ex-pectations” depends on credibility, which can be enhanced through signaling, demonstrat-ing commitment to cooperation, “trial balloons,” attempts at consistent behavior, and a se-ries of small agreements (Dixit and Nalebuff 1991). It is in this area of cautious moves andattempts at creating stable expectations under risk and uncertainty where we hypothesizethat incremental adaptive behavior occurs.

SatisficingA second type of outcome predicted by incremental theory is a satisficing one. If participantsact in a satisficing way, they will incrementally move toward the Nash equilibrium but haltat a suboptimal outcome where both players are minimally accepting of the rewards they re-ceive. Why might this occur? The nature of the risks and uncertainties might be such thatboth players derive a medium payoff with only small risk, yet moving farther toward highermutual payoffs entails larger risks. Due to the uncertainty of keeping a mutual understand-ing intact while moving through the more dangerous larger risks, the players satisfice on aless than optimal outcome.

We maintain that players behave in this incremental and satisficing fashion due to therisk and uncertainty associated with making interdependent decisions. The risks and uncer-

346 Journal of Public Administration Research and Theory

3 We should also note that there is more than one way to use Bayes’s rule in this situation. We have chosen onethat matches up with the overall decision environment. To distinguish it from other possible uses of Bayes’s rule, wecall this “sequential Bayes” or “S-Bayes.”

tainties of misjudging rivals’ or partners’ choices can lead to smaller, cautious steps towardthe Nash equilibrium than Cournot rationality assumptions predict and in some situations afailure to reach the Nash equilibrium at all.

EXPERIMENTAL DESIGN

To systematically examine these ideas about how strategic interdependence influences de-cision making, we run a set of games in a factorial experiment (Oehlert 2000). In each, theexperimental unit is a pair of players who are presented with a decision task based on the du-opoly model (with some alterations). They have a set of preferences but do not know theirpartner’s preferences and, because the experiment is blind, cannot communicate except bymaking simultaneous moves. They play for a fixed number of moves unknown in advanceof play. Each move is recorded.

Table 1 shows an iterative, two-person game (Knott and Miller 1992). There are sev-eral features of this game that simulate the uncertain, complex, and limited information en-vironment that Lindblom and others identify with incremental decision making. First, thegame involves strategies that are continuous variables such that there is a series of best re-sponse (maximizing) reactions by each player to the choices made by the other player. Thematrix is a set of payoffs in which the row chooser, Yolanda, has fifteen alternatives tochoose from. The payoffs are a function of the row choice and the column choice. The gameis also a symmetrical game in which the column chooser, Xavier, has the same matrix ofpayoffs, only transposed. That is, if Xavier picks 3 and Yolanda picks 5, then Xavier gets thesame payoff that Yolanda would get if Xavier picked 5 and she picked 3. In addition, thegame involves simultaneous moves by the players, thus creating a decision environment ofimperfect information. In the plays of the game there is no communication except the pastchoices of the players, neither player knows the other player’s payoff charts, and they areprovided with no information that would provide this knowledge during the play of thegame. The only information provided is the past plays of the other player.4

In this game we are concerned with the strategic interdependence of the players. Inparticular, we examine two factors—uncertainty and risk—both of which should lead play-ers to behave more incrementally, that is, make smaller moves toward the Nash equilib-rium. By uncertainty, we mean that players are given more than one reasonable strategy toplay. By risk, we mean that players face higher negative payoffs for being outside the reac-tion curve. The table 1 game sheet shows the row and column players’ payoffs. The columnplayer’s payoffs, which are indicated in boldface type, are a transpose of the row player’s,which are underscored.5 Note that the players in the game only saw their own payoffs, notthose of their partners. The Nash equilibrium play is (14,14), whereas row 7 has the high-est average value. In addition, we provide an extra column and an extra row (unavailable tothe players) that show the average of the payoffs in each row and column. The Cournot re-action curves in table 1 are indicated by an asterisk. Because of symmetry, the reactioncurves cross at (14,14). The reaction curves shown are derived by simple maximizing be-havior. That is, for any possible choice by one’s partner, the reaction curve indicates thebest possible choice of the row chooser.

We developed four game sheets representing the treatment conditions by crossing the

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 347

4 During play, many players seem to assume the payoffs are the same, however.5 In the experiment, this was accomplished by having all players be row players. Thus one’s partner is naturally thecolumn player.

348Table 1Game Sheet with Both Players’ Payoffs Shown

Column

Row Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Average

1 1.00 1.00* 0.75 0.85 0.85 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.901.00 2.00* 0.00 –1.00 –2.00 –3.00 –4.00 –4.50 –4.75 –5.00 –5.25 –5.50 –5.75 –6.00 –6.25

2 2.00* 1.20 1.00 1.00 1.00 1.00 1.00* 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.081.00* 1.20 1.40 1.60 1.80 2.00 2.20* 0.00 –1.00 –1.25 –1.50 –1.75 –2.00 –2.25 –2.50

3 0.00 1.40 1.25 1.15 1.15 1.10 1.10 1.10* 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.060.75 1.00 1.25 1.50 1.75 1.50 2.20 2.50* 0.00 –1.00 –1.25 –1.50 –1.75 –2.00 –2.25

4 –1.00 1.60 1.50 1.30 1.25 1.20 1.20 1.20 1.20* 1.20 1.20 1.20 1.20 1.20 1.20 1.110.85 1.00 1.15 1.30 1.45 1.45 1.75 1.90 2.05* 0.00 –1.00 –1.25 –1.50 –1.75 –2.00

5 –2.00 1.80 1.75 1.45 1.35 1.30 1.30 1.30 1.30 1.30* 1.30 1.30 1.30 1.30 1.30 1.160.85 1.00 1.15 1.25 1.35 1.40 1.55 1.65 1.75 1.85* 0.00 –1.00 –1.25 –1.50 –1.75

6 –3.00 2.00 2.00 1.50 1.45 1.40 1.40 1.40 1.40 1.40 1.40* 1.40 1.40 1.40 1.40 1.200.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90* 0.00 –1.00 –1.25 –1.50

7 –4.00 2.20* 2.20 1.75 1.55 1.55 1.50 1.50 1.50 1.50 1.50 1.50* 1.50 1.50 1.50 1.250.90 1.00* 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.75 1.80 1.95* 0.00 –1.00 –1.25

8 –4.50 0.00 2.50* 1.90 1.65 1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60* 1.60 1.60 1.170.90 1.00 1.10* 1.20 1.30 1.40 1.50 1.60 1.70 1.75 1.80 1.90 2.50* 0.00 –1.00

9 –4.75 –1.00 0.00 2.05* 1.75 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.70* 1.70 1.000.90 1.00 1.10 1.20* 1.30 1.40 1.50 1.60 1.70 1.80 1.80 1.95 2.00 2.50* 0.80

10 –5.00 –1.25 –1.00 0.00 1.85* 1.80 1.75 1.75 1.80 1.80 1.80 1.80 1.80 1.80* 1.80 0.830.90 1.00 1.10 1.20 1.30* 1.40 1.50 1.60 1.70 1.80 1.80 1.95 2.00 2.50* 1.00

11 –5.25 –1.50 –1.25 –1.00 0.00 1.90* 1.80 1.80 1.80 1.80 1.90 1.90 1.90 1.90* 1.90 0.640.90 1.00 1.10 1.20 1.30 1.40* 1.50 1.60 1.70 1.80 1.80 2.00 2.10 2.50* 1.20

12 –5.50 –1.75 –1.50 –1.25 –1.00 0.00 1.95* 1.90 1.95 1.95 2.00 2.00 2.00 2.00* 2.00 0.450.90 1.00 1.10 1.20 1.30 1.40 1.50* 1.60 1.70 1.80 1.80 2.00 2.10 2.50* 1.40

13 –5.75 –2.00 –1.75 –1.50 –1.25 –1.00 0.00 2.50* 2.00 2.00 2.10 2.10 2.10 2.10* 2.10 0.250.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60* 1.70 1.80 1.80 2.00 2.10 2.50* 1.60

14 –6.00 –2.25 –2.00 –1.75 –1.50 –1.25 –1.00 0.00 2.50* 2.50* 2.50* 2.50* 2.50* 2.50* 2.50* 0.120.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70* 1.80* 1.80* 2.00* 2.10* 2.50* 1.80*

15 –6.25 –2.50 –2.25 –2.00 –1.75 –1.50 –1.25 –1.00 0.80 1.00 1.20 1.40 1.60 1.80* 2.00 –0.580.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.80 2.00 2.10 2.50* 2.00

Column average 0.90 1.08 1.06 1.11 1.16 1.20 1.25 1.17 1.00 0.83 0.64 0.45 0.25 0.12 –0.58

Note: Payoffs are in dollars. Row player’s reaction curve payoffs are underscored. Column player’s reaction curve payoffs are in boldface type.*, Cournot reaction curves; **, Nash equilibrium play and highest average value.

two factors. (These game sheets are provided in appendix 1.) The sheet shown in table 1 andin appendix 1(a) represents the U+R+ condition of high uncertainty and high risk. The highnegative numbers off the reaction curve raise the level of risk to players, depending on theplay of their opponents. The separation of the maximum average value row from the Nashrow introduces alternative strategies for playing the game. In the low uncertainty condition(appendix 1[b]), the row with the highest average value is moved from row 7 to row 14 (theNash row), while maintaining the reaction curve. To do this, we altered the values of cells(14,9) to (14,13) by making them much larger. This preserved the structure outside the reac-tion curve but made row 14’s average value the largest on the table. This change should re-duce the strategic uncertainty for the players by making moves toward the Nash the onlyreasonable strategy for maximizing payoffs. In the low risk case (appendix 1[c]), all negativepayoffs were reduced to 25 percent of their original values (and rounded to the nearest “nice”number). Thus in the U+R– condition, the negative payoffs were 25 percent of the original,but row 7 had the highest average. In the U–R+ condition the exact opposite was true, and inthe U–R– (appendix 1[d]) condition, both the negative payoffs were reduced to 25 percent ofthe original, and the highest average row and the Nash row were the same. Players made se-quential moves on sheets of paper, and the investigators provided updates on each play, to atotal of fourteen plays, though players were not told this number in advance.6

Can we predict how the individuals would play the game? The Cournot solution restson quick updating and confident assumptions about the opponent’s choices. Again, aCournot decision maker adopts the view that his rival’s output is fixed. That is, each personassumes that his opponent will pick in the upcoming period exactly what he picked in theprevious period. Thus if we assume that both people pick outcome 2 in the first period, bothwould pick outcome 7 in the second period and outcome 12 in the third period. Both play-ers would arrive at the Nash equilibrium at outcome 14 at the fourth period. We thus offerthe following hypotheses:

H1a: With low risk and uncertainty, players will follow a Cournot rational-maximizing strategy, which will bring them to the Nash equilibrium in fourplays of the game.

H1b: Sophisticated players will reason through the four plays and choose the Nash equilibrium row (14,14) in the first play of the game.

With high risk and uncertainty, decision makers will more cautiously update their assess-ment of the probability of choices by their opponent. If there is a growing risk in moving to-ward the Nash equilibrium and the row chooser is risk averse, then making confident pre-dictions about the column chooser’s choices may seem too risky. This is especially the caseif there are potential high costs in making an incorrect prediction. (In the game, these costswould be indicated by negative or low positive payoffs.)

Under these conditions, a sequential Bayesian player may prefer to make an assessmentof the average of the prior moves of the other player. As play proceeds, he may prefer to givemore weight to recent moves by his opponent and less weight to earlier moves. Under thiskind of belief structure, there is no single predicted path for all games. In games in whichthe Nash equilibrium row and the MEV row are the same, the sequential Bayesian player is

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 349

6 The experiment was run for fourteen plays to avoid any end-point effects that might be present if the subjectswere to guess the experiment would end at a “nice” number, such as fifteen.

likely to choose the Nash row on the first move. When the MEV row is separate from theNash equilibrium row, the sequential Bayesian player will take longer than the Cournotplayer to reach the Nash equilibrium. Based on calculations of Bayesian probability, weoffer two additional hypotheses:

H2a: Under conditions of high risk and uncertainty, the sequential Bayesian playerswill average probabilities of prior moves by the other player, which will takethem six plays to reach the Nash equilibrium.

H2b: In games in which the MEV row and the Nash equilibrium row are thesame, sequential Bayesian players will choose the Nash equilibrium on thefirst play of the game.

For some incremental players, risky and uncertain conditions may convince them that itwould be better to stick to a safe alternative (a row with no or very few negative numbers).The row choosers may prefer to stick with their original risk-averse or risk-neutral choicesbased on the assumption that any column choice is equally likely. They are reluctant to “up-date” their original uncertain expectations about column choices and will only make small,incremental steps toward the Nash equilibrium.

If Yolanda, for example, behaves according to this view of adaptation, then she will notfollow the Cournot reaction curve or sequential Bayesian updating. On the contrary, if sheand Xavier selected outcome 2 in the first period, she will see that she could be better off bymoving to row 3 or row 4, and that will be sufficient. If Xavier also responds adaptively,they could move from outcome (2,2) to (3,3) to (4,4) and so on. They should still end up atthe Nash equilibrium of (14,14) as long as the process of adaptive responses continues, butit will take perhaps twelve periods, instead of the six periods (at most) in the sequentialBayesian model.

An alternative hypothesis that we consider is that under conditions of high risk, fol-lowing Lustick (1981), decision makers can ill afford to “muddle through” and will seekmore aggressively to find the optimal strategy. With low risk, the surface of choices is flat-ter, and players have the luxury of “wandering around” a bit without fear of major negativeconsequences. With high risk, the surface is more peaked, and decision makers behave in amore rationally maximizing fashion. We offer the following three hypotheses:

H3a: Under conditions of high risk and high uncertainty, players will update theirbeliefs about the probability of their opponent’s set of choices more slowlyand hence take longer than four plays to reach the Nash equilibrium.

H3b: Under conditions of either high risk or high uncertainty, players will updatetheir beliefs about their opponents’ choices more slowly than under condi-tions of low uncertainty and risk, but not as slowly as when both uncertaintyand risk are high.

H3c: Under conditions of high risk, players will update their beliefs quickly aboutopponents’ choices because of the cost of making a “wrong” decision.

Satisficing behavior may result from uncertainty over the most effective strategy for playingthe game. If the decision maker is attracted to safe choices, for example, the row that max-imizes the average value might look attractive as long as he assumes that the column chooser

350 Journal of Public Administration Research and Theory

has some probability of picking any of those columns that result in the high payoffs. Playingthis way, the player can get reasonably high payoffs without incurring the risk of potentialnegative payoffs. In this case, such a player may cling to the highest average payoff row untila string of column choices forces him to concede that those nice columns will not be selected.If both players cling to safe choices, they may take a long time to reach the Nash equilibriumor may not reach it at all. If it becomes riskier to move beyond the safe row, this reinforces thereluctance to change. This combination of risk and strategic uncertainty constitutes a reasonfor a satisficing choice over a maximizing one because both players would be better off ifthey could move to the Nash equilibrium. We offer the following two hypotheses:

H4a: Under conditions of high uncertainty, some players will settle on a satisficinglevel of payoffs to avoid the potential risks and uncertainties of moving farthertoward the Nash equilibrium.

H4b: Under conditions of high risk and uncertainty, some players will not reachthe Nash equilibrium in the duration of the game, thus producing asatisficing outcome.

Thus taking two definitions of optimizing behavior in an interactive game and two simpledefinitions of nonoptimizing adaptive behavior in the same game, we have four alternativepredictions. These predictions are not alternative end-point predictions but rather alternativeprocess predictions. One hypothesis predicts that individuals will follow the reaction curvesto the Nash equilibrium in four periods or less. This is Cournot play. The second hypothesispredicts that individuals will not follow the reaction curves and will reach the Nash equilib-rium in six plays depending on which row maximizes expected value. This is Bayesian play.A third hypothesis predicts that individuals will not follow the reaction curves and will takemuch longer than six periods to reach the Nash equilibrium. This is incremental play. Thefourth hypothesis states that individuals also will not follow the reaction curve but will getstuck at intermediate outcomes and possibly not reach the Nash equilibrium at all duringthe course of the game. This is satisficing play. These alternative paths are represented in fig-ure 1. As a summary of the end-point hypotheses in terms of the number of steps until con-vergence by experimental condition, note that U–R– < U+R– ≤ U–R+ < U+R+.

THE EXPERIMENTAL SETTING

We have chosen a laboratory setting with university students as subjects for playing theseexperimental games. Some experiments in public administration or related fields have beencarried out on a sample from the relevant agency or executive population. Zimmermanand Zysno (1983), for example, used experimental procedures to model the “cognitivehierarchy”—the decision criteria themselves and the way they connect—of how agencyclerks make decisions about the creditworthiness of clients. They tried to ensure that theirstudy had strong external validity by using actual credit clerks as subjects and used simu-lated applications (which served as the experimental task) that were close to the kind facedin actual practice. This task is ideal for experimentation as it is highly structured and re-sembles the real-world problem quite closely.

Experimentation, however, often cannot be carried out with agency employees. Agen-cies typically are protective of their decision processes and may feel that experiments withemployees disrupt work. Executives generally are too busy or have schedules that do not

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 351

easily accommodate experiments. Drawing a sample of experimental subjects from agenciesis also difficult. Agencies are hierarchies with varying decision authority at different levels.Is a representative sample of individuals from each level realistic of the process of how de-cisions are actually made? The organization chart is likely to be out of date or not fully re-flective of actual decision authority in the organization. Informal networks of decision mak-ers may have more power than the formal organization chart might predict.

For these reasons, we have chosen a laboratory setting, with university students as sub-jects, for carrying out the experimental game. This approach is not without its own limita-tions and potential biases, and we would be remiss not to highlight them. Undergraduate stu-dents at large universities may be much more homogeneous than the real population. Thisrestriction of variance may result in the experimenters’ missing a particularly influentialcause in the real world because it is held constant in the sample. If the population fromwhich the sample is drawn is itself unrepresentative of the population at large, randomiza-tion will not control for this bias. For instance, students may perform well above the popu-lation average at tasks such as multiple-choice tests but may lack sufficient experience to un-derstand certain kinds of complex decisions and thus perform below the population averageon such tasks. In general, agency heads are much more experienced strategic thinkers thanstudents; indeed, a J. Edgar Hoover or a Donald Rumsfeld is a world-class strategist.

The laboratory setting itself also can be highly biasing. Experimenters such as Quat-trone and Tversky (1984) or Milgram (1974) make use of “white coat” authority in thetreatment design itself, but more often it is a source of nuisance variation as subjects re-spond to the unwanted and uninteresting treatment of simply being in a laboratory. For in-stance, in studying work-group tasks in a laboratory experiment, the problem is that realwork groups are rarely as ad hoc and short lived as experimental work groups, and realtasks are usually much more salient and well understood than experimental ones.

352 Journal of Public Administration Research and Theory

16

14

12

10

8

6

4

2

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14

CournotS-BayesIncrementalSatisficing

Round

Play

Figure 1Hypothetical Play Curves: Cournot, S-Bayes, Incremental, Satisficing

In this experiment we recruited students from three distinct sources so as to maximizethe heterogeneity as much as possible, at least given that all were from one university. Stu-dents came from introductory political science and mathematics courses, both of whichdraw quite broadly from different majors on campus. In addition, we recruited subjects fromone of the campus sports clubs, which also has a broad range of majors. In all there wereeighty subjects of mixed race and major, with ten pairs per cell. There were thirty-six menand forty-four women. So as not to confound treatment with a particular run of the experi-ment, each time we ran the experiment we included one of each treatment type. A particu-lar run of the experiment took about an hour.

During the experiment the subjects were read the following instructions, which alsoappeared at the top of their game sheets (reproduced in appendix 1):

In this experiment, you will play a game with an anonymous partner. Your task will be tochoose a row to play. Your partner will also choose a play at the same time. Depending onwhat you both choose, you will receive the payoff (in dollars) indicated in the box on thegame matrix below. You pick rows. Depending on what your partner plays, you receive thevalue in the box. For instance, if you picked 1 and your partner picked 1, you would receive$1.00. After each play, you will get to see your partner’s play history.

They were also explicitly told that their partner’s sheet could be different from theirs andthat they would be informed only of their partner’s past play during the course of the game.There was no recruitment fee; subjects’ earnings came entirely from game play, though weset a minimum payout of $5 and, because of the budget, a maximum payout of $25.

ANALYSIS

The primary data gathered in this experiment—the play sequences—have two aggregate as-pects that we focus on here. Then we examine some of the deviant play sequences. The firstaspect is that of the path taken to equilibrium, and the second aspect is the equilibriumreached. Are there systematic differences in the path taken to equilibrium between treatmentgroups? Are there systematic differences in the number of steps taken to equilibrium betweenthe treatment groups? If the Cournot duopoly model is correct, the treatments should have nosystematic effect, and the plays should follow the predicted Cournot path, at least on average.If, on the other hand, the treatments do have an effect, systematic differences should be ob-served. In fact, systematic differences are observed, as we show in the following discussion,though they are not entirely consistent with the hypotheses we set out.

Figure 2 shows the median play by treatment group as well as the path predicted by theCournot duopoly game. Clearly the overall paths in all treatment groups go from low rownumbers toward row 14, which is the basic shape predicted by the Cournot duopoly game.However, some systematic differences are apparent. In each treatment group, subjects werewilling to start with a substantially higher row play than one would expect, with the U–conditions on average starting higher than the U+ conditions. The U+R+ and U+R– con-ditions were closer to the Cournot play than the U–R+ and U–R– conditions, at least in theaggregate, and converged at the predicted stage. They greatly resemble the sequentialBayesian solution, both in terms of where they start and how long it takes to converge. TheU–R+ and U–R– conditions, however, did not converge on average until several steps laterthan the Cournot prediction. The sequential Bayesian solution prediction is completely in-

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 353

accurate—players should converge immediately. In fact, the U–R+ median pulls away fromthe Cournot play of 14 in round twelve, as can be seen by the slight dip in the figure. Thusat a gross level, the Cournot duopoly model fits, particularly for the U+R+ and U+R– con-ditions, but there are systematic differences due to the treatments. At this point it seems thatU dominates R because the curves differ primarily on those treatments. We will examinethese systematic differences more fully in subsequent paragraphs and demonstrate that thisclaim is supported by the experimental evidence.

Visually, it seems most of the difference is clearly in the first few rounds of play, and themain line of separation seems to be along the U+ versus U– lines. To formally examine theearly plays, we define two dummy variables. U = 1 if we have treatment groups U+R+ orU+R– and 0 otherwise. R = 1 if we have treatment groups U+R+ or U–R+ and 0 otherwise.Using the nonparametric7 Mann-Whitney test for difference of location between two distri-butions, we see that the U factor clearly separates the distributions in the first round of playand is at least suggestive in the second round of play, with the z-statistic slightly less than 2in absolute value. All difference is gone by the third round of play. In contrast, there are nostatistically significant differences between the two levels of R. Table 2 shows these results.8

354 Journal of Public Administration Research and Theory

7 Because the variables considered in this analysis are row selections, they are best treated as ordinal, not interval.Thus the analysis undertaken here will be entirely based on nonparametric or semiparametric methods. In general, therelative efficiency of nonparametric methods is somewhat lower than parametric alternatives when the assumptionsbehind the parametric methods are true. However, nonparametric methods require weaker assumptions and thus leadto more robust inferences. When all goes well conclusions are similar, but nonparametrics provide useful protectionagainst method artifacts.8 One criticism that might be leveled here is that the data are structured into pairs of players, and the observationsare thus not independent. Note however that in the first round of play they are in fact independent because the playershave no information about past play and are thus responding to the stimulus alone. The second round has some corre-lation due to last round’s information, but it is still relatively little.

14

12

10

8

6

4

2

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

U+R+

U-R+

U+R-

U-R-

Cournot

Round

Med

ian

Play

Figure 2Median Play by Treatment Group

Next we will consider the convergence of pairs of players to the Nash equilibrium.Figure 3 shows by treatment group the round at which a percentage of the subjects con-verged to the Nash equilibrium. A notable feature is that equilibrium is not reached by asubstantial proportion of players, and when it is, the players take longer than the Cournot so-lution would predict. A Cournot pair of players should converge by round four. As we sawearlier, the median player has in fact converged by round five in some treatment conditions.However, this fact says nothing about pairs of players. If players played according to theCournot strategy, we should expect pairs to converge to the Nash equilibrium near roundfour. The U+R+ and U+R– conditions converge somewhat later than that, with the medianround to convergence being round seven in both cases. This is consistent with sequentialBayesian play on these games and is not a large deviation from that predicted by the Cournot

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 355

Table 2Medians by Factor in First Five Rounds

Round 1 Round 2 Round 3 Round 4 Round 5

Cournot play 2 7 12 14 14U = 0 Median play 8 11 10.5 12 12.5U = 1 Median play 4.5 8 10.5 11 14Z –2.269 –1.914 –.054 –.256 –1.016R = 0 Median play 7 10 9.5 11 13.5R = 1 Median play 7 10 12 9.5 14Z –.083 –.200 –.906 –.803 –.890

Note: U = 1 if we have treatment groups U+R+ or U+R– and 0 otherwise. R = 1 if we have treatment groups U+R+ or U–R+and 0 otherwise.

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0%

1 2 3 4 5 6 7 8 9 10 11 12 13 14

U+R+

U-R+

U+R-

U-R-

Round

% C

onve

rged

to th

e N

ash

Figure 3Percent Nash Convergence at Rounds by Treatment

strategy. However, the median round to convergence in the U–R+ and U–R– conditions isround thirteen and censored (i.e., greater than fourteen), respectively. None of the pairs inthese conditions converged by round four or before. Evidently players do not follow theCournot game but instead converge more incrementally, and the chance a pair will deviatefrom the Cournot solution is increased by the U manipulation. The sequential Bayesian so-lution fits reasonably well on the two U+ games but not at all on the U– games.

There is a further subtlety about this figure, given that a substantial proportion of thepairs did not reach equilibrium. Because the game was played for only fourteen rounds, anumber of the pairs are censored before convergence.9 As such, the figure combines bothquantitative information—“At what round do pairs converge?”—and qualitative informa-tion—“Do they converge at all in the scope of the experiment?” Clearly there are system-atic differences between the curves, with the U– conditions taking substantially longer toreach convergence than the U+ conditions. A substantially larger proportion of the U–R–condition pairs never reached convergence at all, and many of the U–R+ condition pairsonly converged in the last round or two of play. In contrast, the R condition seems to makeno difference at all, at least in what we observed.10

To examine more formally the differences between treatments in the duration taken toconvergence to the Nash equilibrium, taking censoring into account, we used a survivalanalysis model, the Cox Proportional Hazard (CPH) model (Kalbfleisch and Prentice 1980).The Cox model is a semiparametric duration model that focuses on comparing the effect ofcovariates on the survival curve. In this case survival means “has not converged to the Nashequilibrium.” The model computes a baseline hazard—basically a nuisance variable in thiscase—nonparametrically and models the effect of covariates multiplicatively, thus the term“proportional hazard.” Ordinary regression analysis has no means to deal with the partiallyinformative observations of the censored cases except in an ad hoc fashion, but the Coxmodel uses the information from the censored cases appropriately.

The model shown in table 3 uses the two dummy variables, U and R, defined previously.They are entered additively into the CPH equation.11 To interpret a coefficient b, one examinesthe natural exponential of the regression coefficient, exp(�). A value near 1 indicates that thereis no proportional change from the baseline hazard function. In this case, U = 1 increases the

356 Journal of Public Administration Research and Theory

9 It is unclear whether nonconverging pairs would ever converge to the Nash equilibrium if given sufficient roundsto play. In most cases we believe they would. In a few other cases the players converged to a regular pattern of play—for instance (7,7) or alternating between (12,14) and (14,12)—but it was not the Nash equilibrium.10 However, because the experiment is censored before all pairs converge, there is no way of knowing when the U-R- condition would have converged, though it is interesting to note that the U-R+ condition has a rush of conver-sion near the end of the game. Without further plays of the game, we cannot know the outcome. Unfortunately, subjects’ attention spans are substantially taxed even by fourteen plays, and it would be very difficult to continue theexperiment much beyond that point while maintaining validity.11 We fit the model with U+ and R+ interacted, but the coefficient is not significant and the entire model becomesinsignificant. It is not shown here.

Table 3Cox PH Model Results

Factor � SE p-Value exp(�)

U .833 .400 .037 2.301R .325 .398 .414 1.384

Note: U = 1 if we have treatment groups U+R+ or U+R– and 0 otherwise. R = 1 if we have treatment groups U+R+ or U–R+and 0 otherwise.

proportional hazard of convergence by roughly a factor of 2. That is, taking censoring intoaccount, when U = 1, pairs converge to the equilibrium a bit more than twice as fast as whenU = 0. When R = 1 increases the hazard of convergence slightly, but it is not statistically signi-ficant. This formal test supports figure 3. In summary, the aggregate results show that, contraryto our previous end-point hypothesis, U+R+ = U+R– < U–R+ ≤ U–R–.

Finally, we consider some of the deviant play sequences in more detail, as they bothprovide more context and shed some light on the reasons why play differed from the givenhypotheses. One of our hypotheses was that satisficing play might be observed with a pairof players converging to a suboptimum. Though satisficing was not supported in the aggregateresults, a few pairs did show evidence of it. One particularly striking pair rapidly convergedto (7,7) and never moved from there, and some others eventually settled on a satisficing-typeequilibrium. One player commented in the questionnaire handed out at the end of the ex-periment, “My partner didn’t change his answer much as it was easier.” Similarly, in gameU+R+, two players followed small incremental steps to reach convergence at the Nash equi-librium on the seventh play of the game. One player started out in the Nash row but bouncedto lower rows to accommodate the choices of the other player. This player started in row 7and incrementally moved over the next seven plays to row 14.

One particularly clever pair of players in the U–R– condition converged to a two-steptrading cycle of (12,14) and (14,12), which would clearly be superior to the Nash play inpayoff. This trading cycle points to an interesting issue that explains the reversal in the end-point hypotheses. The U– treatment was implemented by moving the safe row with thehighest average value so that it was consistent with the Nash row. This was done by puttingsome large positive values on the reaction curve but not on the Nash equilibrium. In theCournot story these values should make no difference at all. However, as is evident from theaggregate results presented, subjects responded quite strongly to them. Unlike the standardCournot duopoly, subjects in this game did not know the reaction curve but instead had toinfer it, so perhaps this is not so surprising. We speculate that subjects were attracted to thelarge numbers and thus tended to pull away from the Nash equilibrium in an attempt to luretheir partners to play a lower value. If this strategy was successful, as in the case of theplayers who set up the trading cycle, it was possible for both to do better than simply play-ing the Nash, similar perhaps to logrolling in legislatures. Although the Nash equilibrium inthis game is Pareto-efficient in any one play, play here is sequential. That said, it is not ter-ribly surprising that it is very difficult for players actually to manage this trading cycle givenhow little they know. In fact, one of the earliest experiments in game theory—discussedextensively in Poundstone (1992, 106–21)—was performed by Merrill Flood and MelvinDrescher, using economists Armen Alchian and John Williams as subjects. It exhibits ex-actly this pattern, though only after a substantial learning time (dozens of plays).

More commonly, a number of pairs exhibited “wandering” behavior in which it seemedthat partners did not understand each other’s motivations. In such cases, usually one partnerclearly tried to play something more or less like the Cournot strategy, but the other player didnot understand it and thus did not respond in kind (which lead to substantial monetary losses),so the other partner stopped playing Cournot. Sometimes a pair of players would “dance”with each other one round out of sequence. As mentioned, we asked the subjects a battery ofopen-ended questions after the experiment, including one about whether they thought theirpartners understood the game. The responses were generally consistent with the lack of co-ordination interpretation. In one particularly bizarre case, one player seriously misunder-stood the game itself and played a pattern that was highly regular but totally unrelated to the

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 357

payoff structure (it was tent-shaped, starting low, moving to 14, and then moving back downagain). His partner noted in the open-ended response that he was totally confused by the part-ner’s play, as he had expected the partner to respond to the payoffs in the game matrix.

In games U–R+ and U–R–, we found a pattern of negative reinforcement and distrustbetween two pairs of players. In both cases, one player would start with a high row and theother would pick a middle row. In the next play, their choices would be reversed. This in-verse pattern caused each of them to lose money and set them on a pattern of interaction thatprevented convergence to the Nash equilibrium. One player commented about her partner,“She countered me with numbers opposite mine, or a set of numbers to make larger amountsof money.” The other player observed of her partner, “[Her choices] seemed random, espe-cially after the first seven or so plays.”

The other pattern that emerged from U–R+ and U–R– games was a mismatch betweenone player who moved to the Nash row quickly while the other player bounced up and downbetween high and low rows, not sure of a consistent strategy. One player who consistentlypicked row 14 commented about the erratic play of her partner, “[She] seemed quite unco-operative. [My partner] was for greed and had a stubborn reluctance to grow as a team.”About her own play, she said, “I was trying to accomplish cooperation, so that we mightboth get the most amount of money.”

Another observable incremental pattern that occurred in U–R– was incremental con-vergence that took several plays of the game. One player in particular played a very cautiousgame, starting with row 2, then 4, 5, 6, 12, 13, 15, 14, 12, 10, and finally converging backat 14. Her partner had sought to move to row 14 by the third play of the game but moved tolower rows to avoid the costs incurred by the partner’s incremental approach. In some cases,therefore, a consistent incrementalist can induce more incremental play in a partner.

DISCUSSION

The concept of incremental adaptation entered the social sciences literature because empir-ical observations of behavior did not fit with a fully rational approach to decision making.Early explanations of incrementalism included an ambiguous mixture of cognitive limitsby decision makers faced with complex problems and the politics of decentralized interac-tion. For the most part, however, the public administration, budgetary, and organization lit-erature has emphasized the cognitive limits explanation for incrementalism. Indeed, Model2 in Graham Allison’s book The Cuban Missile Crisis (1971) is based on the March andSimon (1958) notion of how simplified organizational routines serve as heuristics to copewith the complexity of decisions.

The development of institutional economics and game theory has focused much moreon the effects of strategic interdependence under limited information and uncertainty on thecollective effect of individual choices. In this article, therefore, we have sought to induce in-cremental behavior in players faced with strategic interdependence through a two-person, simultaneous-play game. We have incorporated risk and uncertainty into the interdepend-ence of the game situation while holding constant the complexity of the choice set and thesequence of play. Our results from this game show that strategic interdependence in thegame situation does induce incremental behavior along the lines predicted. After fourteenplays of the game, of the forty pairs in the experiment, 35 percent had not reached the Nashequilibrium. Cournot behavior, defined as convergence before or at four plays of the game,was exhibited in only 30 percent of the pairs.

358 Journal of Public Administration Research and Theory

On the other hand, the aggregate results for number of plays to reach convergence to theNash equilibrium showed a predictable, although slow, Cournot pattern. This overall medianpattern of convergence occurred despite the fact that individual pairs of players varied in sev-eral directions in each of the games. Hence we conclude from these results that although somedecision makers exhibited incremental behavior in these complex decision situations, the over-all average follows a pattern of choice relatively close to the one predicted by the Cournotstrategy. This pattern is not, however, necessarily exhibited within a given pair of players.

The predictions about the effects of specific dimensions of interdependence were notborne out. The results did reveal examples of classic incremental adaptation and satisficing be-havior. A few players settled on row 7 as a medium strategy that gave them satisfactory pay-offs without the greater risks of moving toward the Nash equilibrium. More players movedcautiously from low-numbered rows in a stepwise fashion over several plays to reach theNash equilibrium row. Players also found themselves in an unstable pattern of inverse choicesthat reinforced mistrust and lack of credibility, preventing them from reaching the Nash equi-librium at all. In some cases, steady play by one player at the Nash row slowly induced theother player to conform; in other cases, steady play of this sort did not produce this result.

If the overall pattern approximated the Cournot solution, how much importance shouldwe place on the incremental subpatterns and the unstable inverse patterns that occurred insome cases? One answer is that we should not place emphasis here because on average itseems to all work out according to Cournot rational theory. Perhaps an analogy could bemade to the economy, where under conditions of competition, the overall economy per-forms efficiently even if individual firms make mistakes and go out of business. Of course,the number of players in these limited experiments is far too small for this kind of analogyto work, but the average success of an approximate Cournot strategy suggests the power ofadaptive Cournot solutions.

The literature often asserts that incrementalism is not rational but does not really explainthat claim. However, we ask whether or not it might be rational in some conditions to chooseincrementally. This gets into the matter of the level of risk aversion or acceptance in a givenplayer as well as the cost of information. A pessimistic risk-averse player values certaintyover potential gain, whereas an optimistic risk-acceptant player values potential gains overcertainty. These are different preference structures. Rational choice theory does not claim tochoose between preference structures but instead takes them as given; in other words, it doesnot purport to “account for tastes.” In this respect, a Cournot player represents an optimisticplayer who confidently (arguably too confidently, if the sequential Bayesian rule is the stan-dard of rationality) updates information. In addition, some sorts of incremental play couldwell be rational in the sense that they would be consistent with a player’s preferences and be-liefs. Similarly, if information is costly, incrementalism might be rational, depending on howplayers value it. That said, incremental play would clearly be irrational if the player madechoices inconsistent with preferences and beliefs. In our view, claiming that incremental playis irrational excessively focuses on retrospective assessments (i.e., “Monday morning quar-terbacking”). But players in a limited information game do not have the luxury of such as-sessments because they do not possess the information available retrospectively.

Another way of looking at these results is in terms of the lessons to be learned aboutstrategic interaction from the behavior of adaptive incrementalism. In some instances a sub-pattern of instability persisted for the entire play of the game, preventing a convergence tothe Nash equilibrium. From these patterns we can learn how negative feedback causes insta-bility and the possible ways to break out of inverse reactions. In other instances, subpatterns

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 359

of incremental adaptation led to several plays before the equilibrium point was achieved.From these patterns we can learn how incremental moves in the right direction can inducemutual gain and even possibly prevent inverse instability from occurring. A few playerssettled on row 7 as a medium strategy that gave them satisfactory payoffs for the entire playof the game without the greater risks of moving toward the Nash equilibrium. From thispattern we can learn how initial instability and inverse decisions can lead to suboptimal pat-terns that persist due to prior negative interactions.

What causes the mismatch remains to be seen, but we speculate that it is the result oftwo effects, one relating to preferences and the other to cognitive ability. It is possible thatthe mismatch represents variation in acceptance of risk. It may be that when an optimisticplayer willing to take chances partners with a pessimistic player who is not, the playershave difficulty forming stable mutual expectations because the optimistic player, by playingaggressively, ends up being inadvertently punished by the more conservative player. Mis-match may also be caused by cognitive ability differences between players in a similar fash-ion. It was quite clear from the free response items that some players had much better un-derstanding of the game than others. A player who manages to solve the game quickly andplays a high row quickly might, however, be punished by a partner who does not.

These observations show that it is too limiting to identify incrementalism only withproblem complexity and limitations in rationality for dealing with it. In Allison’s models ofdecision making (1971), for example, Model 2 is based squarely on the incremental litera-ture in organization theory and decision making. Model 2, however, ignores the effect ofstrategic interdependence on incremental behavior (Bendor and Hammond 1992). Allisonincludes interdependence in Model 3, the bureaucratic politics model, through his refer-ences to Lindblom (1965), although there is still a strong emphasis on limited analysis.Under some conditions of decentralized interaction, given limited information, sequentialplay, and low risk acceptance, decision makers will behave incrementally in Model 3. In ad-dition, more knowledge and analysis about the decision problem or faster computationalability cannot eliminate this kind of incremental behavior. Mismatches in risk preferences,moreover, can lead to unstable interactions, but the players are not necessarily behaving ir-rationally. Interestingly, a player who does not understand the problem playing against anopponent who does can still produce this kind of unstable interaction.

In Model 1, Allison allows for variants in the classical-rational model to include se-quential play and the preference functions of a leadership clique, or “hawks and doves,” asdecision makers. Such variants can produce Cournot behavior. Less confident assumptionsabout risk preferences and other preference functions of the opponent can produce incre-mental behavior. Given the mutual effect of interdependence on players’ choices (Tsebelis1989), cautious updating of assumptions is not necessarily irrational. Consequently, thesemodels can overlap in concept and in predicted behavior without abandoning rationality as-sumptions. Recognition of the effects of strategic interdependence on decision making, there-fore, may help integrate rational explanations of the patterns of sequential interaction amongcommittees, interest groups, and agencies, as well as among foreign policy decision makers.

CONCLUSION

In sum, the game shows not only the explanatory power of the Cournot and sequentialBayesian solutions but also the influence of strategic interdependence on inducing otherpatterns of behavior. In addition, it points to serious anomalies in the standard theoretical

360 Journal of Public Administration Research and Theory

explanations in some cases, which indicates that current theory is in need of further elabo-ration. To test the specific hypotheses of how dimensions of interdependence influence be-havior would require additional players over more runs of the game. Because some playersnever converged, adding more plays may also give insight into the persistence of instabil-ity and incremental patterns. Unfortunately, our experience running this as an experimentalgame shows that it would be difficult, perhaps impossible, to run the experiment for moreplays. Developing a computer simulation of the game would permit both more players anda larger number of plays. In addition, it would also allow explicit control over (virtual)players with mismatched strategies (e.g., one playing a Cournot strategy while the othersatisfices, something that cannot be done in the laboratory at all). The extent to which play-ers do not reinforce each other’s expectations may be an important predictor of conver-gence. Finally, it would be possible to formalize the prior probabilities of the players, ex-amining the effects of differing belief structures or schemes of updating.

The results of our experiment do point to one central unanswered question in the incre-mentalism literature: How would one empirically recognize incrementalism? We endeav-ored to create a situation in which there was a clearly defined, rational alternative throughwhich to determine if players would play incrementally. Unfortunately, in most real world set-tings the objective functions of players are not known, and much pertinent information ishidden from view. In fact, a large part of the task that decision makers in a duopoly situationface is to find out what their partner’s objective function is. Thus in some circumstances,rational play might well appear—or indeed even be—incremental, given certain preferenceand belief structures.

APPENDIX 1

Game Sheets Given to Subjects with Treatments Noted

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 361

362 Journal of Public Administration Research and Theory

APPENDIX 2

Derivation of Updating Equations and the Sequential Bayesian Strategy

In the context of these games, Bayesian updating and Cournot adjustment can be placedwithin the same mathematical framework using updating equations to alter the probabilitydistribution representing a player’s belief over her opponent’s possible plays. (It is assumedthat the player knows what plays are possible, but does not know the payoffs his or her op-ponent attaches to them.) Because of the information structure of the games described ear-lier, a number of useful simplifications naturally arise.

Let t = 0,1., . . . , tmax, be the round of play; p be a vector of probabilities over the op-ponent’s possible plays at round r; o be the observation vector, which has zeros everywhereexcept a 1 in the position of the play selected by the opponent (making it a degenerate prob-ability distribution); B is the player’s payoff matrix; and u Œ (0,1) an updating equationweight parameter, with larger values indicating more weight given to prior information. Ifwe adopt an expected value maximizing strategy, the player should choose the row thatmaximizes the expected value at round t, et, given by the vector product et = B pt. All weneed to project probabilities is an initial probability assignment and the observations of thelast round’s play. Assume that p0, the prior probability, is given. Then pt+1 = u pt + (1 – u) ot,which is simply an invocation of the theorem of total probability. It is not difficult to back-substitute to replace all pt terms for t > 0 with p0, the prior, and sums of ot-1, ot-2, . . . , o1

alone, weighted by powers of u and (1 – u).In the games given here it is rational to choose an uninformative (i.e., uniform) prior

because nothing is known about the opponent’s payoffs aside from the number of availableplays. The weighting parameter u has a large impact on the updating of probability as it de-termines the value of new information. In terms of the theorem of total probability, it is thesubjective probability assigned by the player to the past states and the observed play. For u = 0.5, it is the player. For u < 0.5, new information is weighted relatively more than old,whereas for u > 0.5, old information is weighted relatively more than new. If u = 0, the up-

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 363

dating equations give Cournot adjustment, because the player in this case assigns no prob-ability to past states whatsoever. In contrast, if u = 1, the player does not update. The priordecays geometrically in u t, so an informative prior will have relatively limited effect aftera few rounds of play, assuming that u is at least moderate in size.

In a symmetric game such as the ones used in this experiment—which, recall, wereconstructed to all have the same Cournot path—we can calculate the sequential Bayesianpath without too much trouble if we assume a uniform prior and the same weighting pa-rameter for both players, though the paths differ by game sheets. Games U+R+ and U+R–have paths (7, 8, 12, 12, 13, 14 . . . ), predicting convergence by round six, not dissimilar tothe Cournot path but a bit more gradual and with a higher starting value. In contrast, gamesU–R+ and U–R– should both converge in the first round of play because row 14 has thehighest expected value under uniformity. Other solutions are straightforward to calculategiven the updating equations and different prior probability vectors.

REFERENCES

Allison, Graham T. 1971. Essence of decision: Explaining the Cuban missile crisis. Boston: Little,Brown.

Bendor, Jonathan. 1995. A model of muddling through. American Political Science Review 89, no. 4:819–40.

Bendor, Jonathan, and Terry Moe. 1985. An adaptive model of bureaucratic politics. American PoliticalScience Review 79, no. 3:755–74.

Bendor, Jonathan, and Thomas H. Hammond. 1992. Rethinking Allison’s models. American PoliticalScience Review 86, no. 2:301–22.

Coase, Ronald. 1937. The nature of the firm. Economica 4:386–405.Crecine, John P. 1969. Governmental problem solving: A computer simulation of municipal budgeting.

Chicago: Rand McNally.Davis, Otto A., M. A. H. Dempster, and Aaron Wildavsky. 1966. A theory of the budgetary process.

American Political Science Review 60, no. 3:529–47.Dixit, Avinash, and Barry Nalebuff. 1991. Thinking strategically: The competitive edge in business,

politics, and everyday life. New York: W. W. Norton.Dixit, Avinash, and Susan Skeath. 1999. Games of strategy. New York: W. W. Norton.Fouraker, Lawrence E., and Sidney Siegel. 1963. Bargaining behavior. New York: McGraw-Hill.Gintis, Herbert. 2000. Game theory evolving. Princeton, N.J.: Princeton University Press.Henderson, James M., and Richard E. Quandt. 1980. Microeconomic theory: A mathematical approach.

3d ed. New York: McGraw-Hill.Jones, Bryan D. 1999. Bounded rationality. Annual Review of Political Science 2:297–321.Kalbfleisch, J. D., and R. L. Prentice. 1980. The statistical analysis of failure time data. New York:

Wiley.Knott, Jack H., and Gary Miller. 1992. The dynamics of convergence and cooperation: Rational choice

and adaptive incrementalism. Unpublished ms. Michigan State University.Lindblom, Charles. 1959. The science of “muddling through.” Public Administration Review 19, no.

1:78–88.Lindblom, Charles. 1965. The intelligence of democracy. New York: Free Press.Lustick, Ian S. 1981. Explaining the variable utility of disjointed incrementalism. American Political

Science Review 74, no. 3:342–53.March, James G., and Herbert A. Simon. 1958. Organizations. New York: Wiley.March, James G., and Johan P. Olson. 1976. Ambiguity and choice in organizations. Bergen, Germany:

Universitetsforlaget.McCubbins, Mathew D., and Thomas Schwartz. 1984. Congressional oversight overlooked: Police

patrols vs. fire alarms. American Journal of Political Science 28, no. 1:165–79.

364 Journal of Public Administration Research and Theory

Milgram, Stanley. 1974. Obedience to authority. New York: Harper & Row.Oehlert, Gary W. 2000. A first course in design and analysis of experiments. New York: W. H. Freeman.Poundstone, William. 1992. Prisoner’s dilemma. New York: Doubleday.Pindyck, Robert S., and Daniel L. Rubinfeld. 2001. Microeconomics. 5th ed. New York: Prentice Hall.Quattrone, George A., and Amos Tversky. 1984. Causal versus diagnostic contingencies: On self-

deception and on the voter’s illusion. Journal of Personality and Social Psychology 46, no. 2:237–48.

Rourke, Francis. 1984. Bureaucracy, politics, and public policy. Boston: Little, Brown.Schelling, Thomas. 1960. The strategy of conflict. Cambridge, Mass.: Harvard University Press.Shepsle, Kenneth A. 1979. Institutional arrangements and equilibrium in multidimensional voting

models. American Journal of Political Science 23, no. 1: 27–59.Simon, Herbert A. 1957. Models of man. New York: Wiley.Steinbrunner, John D. 1974. The cybernetic theory of decision. Princeton, N.J.: Princeton University

Press.Tsebelis, George 1989. The abuse of probability in political analysis: The Robinson Crusoe fallacy.

American Political Science Review 83, no. 1:77–91.Wildavsky, Aaron. 1964. The politics of the budgetary process. Boston: Little, Brown.Williamson, Oliver. 1985. The economic institutions of capitalism. New York: Free Press.Zimmerman, H.-J., and P. Zysno. 1983. Decisions and evaluations by hierarchical aggregation of

information. Fuzzy Sets and Systems 10:243–60.

Knott, Miller, and Verkuilen Adaptive Incrementalism and Complexity 365