Games and Economic Behavior 49 (2004) 374–400 www.elsevier.com/locate/geb

The value of commitment in Stackelberg games with observation costs ✩ Felix Várdy International Monetary Fund, 700 19th Street NW, Washington, DC 20431, USA Received 16 October 2001 Available online 14 November 2003

Abstract We study Stackelberg games in which the follower faces a cost for observing the leader’s action. We show that, irrespective of the size of the cost, the leader’s value of commitment is lost completely in all pure-strategy equilibria. However, there also exists a mixed-strategy equilibrium that fully preserves the first-mover advantage. In this type of equilibrium, the probability that the follower looks at the leader’s action is independent of the cost of looking.  2003 Elsevier Inc. All rights reserved. JEL classification: C72; D83; L13 Keywords: Stackelberg duopoly; Observation cost; Value of commitment; First-mover advantage; Leader–follower game

1. Introduction It is well known that a firm can sometimes gain an advantage by committing to an action ahead of its rivals. Indeed, the familiar Stackelberg quantity-setting duopoly model is a prominent example (Von Stackelberg, 1934). Here, a leader firm obtains an advantage by committing to produce a large quantity of some homogeneous good. The follower, upon observing the leader’s choice, then optimally decides to produce less of the good. The leader thereby gains market share and profit at the expense of its rival. ✩

The views presented in this paper are those of the author and do not necessarily reflect the position of the International Monetary Fund. E-mail address: [email protected]. 0899-8256/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2003.07.003

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Given the prominent role that commitment plays in applied economic theory and in the real world, the robustness of the value of commitment is, of course, quite important.1 However, in a thought-provoking paper Bagwell (1995) has shown that the value of commitment may in fact be extremely fragile. More specifically, he studies leader–follower games where, with small but positive probability, the follower receives the wrong signal as to the leader’s action. The pure-strategy Nash equilibrium outcomes of such a ‘noisy leader game’ turn out to be equal to the pure-strategy Nash equilibrium outcomes of the simultaneous-move version of the game. In other words, the leader’s value of commitment may not be robust at all to noise in the communication technology.2 Van Damme and Hurkens (vDH, 1997) partly salvage the value of commitment in such a world by showing that noisy leader games always have a mixed-strategy equilibrium in which the value of commitment is preserved asymptotically when the noise vanishes. vDH call such an equilibrium a ‘noisy Stackelberg equilibrium’ and develop an equilibrium selection theory according to which noisy Stackelberg equilibria are selected. Finally, in a continuous-action version of the noisy leader game, Maggi (1999) shows that the value of commitment can be restored when there is private information on the part of the leader. In this paper we take a somewhat different approach. We show that, in order to undermine the value of commitment, the introduction of exogenous uncertainty through noisy communication is in fact quite unnecessary. The value of commitment can just as well be compromised by uncertainty that arises strictly endogenously, when monitoring the leader’s action involves some small cost. More specifically, we consider a broad class of leader–follower games in which the follower’s decision whether to observe the leader’s action is determined endogenously. The sequence of moves is as follows. First, the leader takes an action. Then, the follower gets to observe what the leader has done if and only if he expends an amount ε. Finally, the follower takes an action of his own. We shall refer to this class of games as ‘costly leader games’ and study the properties of their equilibria. Our main findings are as follows: 1. In any pure-strategy subgame perfect equilibrium of a costly leader game, the value of commitment is lost completely, no matter how small the follower’s cost of becoming informed (Proposition 2). 2. For sufficiently small costs of becoming informed, there exists a mixed-strategy equilibrium of the costly leader game that perfectly preserves the leader’s first-mover advantage (Corollary 8 and Proposition 10). 3. In all equilibria that preserve the leader’s first-mover advantage, the probability that the follower chooses to observe the leader’s action is independent of the cost of observation (Proposition 9). 1 In this paper, we use the terms ‘value of commitment’ and ‘first-mover advantage’ interchangeably. Both terms refer to the extra equilibrium payoff the leader gets from moving first, as compared to his equilibrium payoff when the players move simultaneously. 2 In the context of a 2 × 2 example, Bagwell also describes the set of mixed-strategy equilibria. In one of these equilibria the value of commitment is preserved asymptotically when the noise vanishes.

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Some of the differences between our costly leader model and Bagwell’s noisy leader model are as follows. First, the absence of underlying uncertainty implies that we can use subgame perfect equilibrium as our solution concept, thereby remaining parallel to the benchmark Stackelberg model. It also means that we do not have to rely on the non-moving-support assumption for our results.3 Second, in all noisy Stackelberg equilibria of the costly leader game, the leader’s value of commitment is preserved completely, while in the noisy leader game the leader’s value of commitment is preserved only asymptotically. Thus, in the costly leader game, the discrepancy between the predictions of the two types of equilibria is sharpened: Purestrategy equilibria predict the complete loss of the value of commitment, while mixedstrategy Stackelberg equilibria predict its complete preservation. Third, equilibrium play in the costly leader model is more readily tested empirically than in the noisy leader model. Key is that, in the costly leader game, the presence of characteristic and observable implications of noisy Stackelberg equilibria does not depend on the occurrence of small probability events. More specifically, in all noisy Stackelberg equilibria of the costly leader game, the follower’s probability of choosing to observe the leader’s action is independent of the cost of observation. This independence property allows us to always clearly distinguish sophisticated noisy Stackelberg equilibrium play from heuristic, non-equilibrium Stackelberg play, even for very small values of the (cost-)parameter. In noisy leader games, by contrast, noisy Stackelberg equilibrium and naive, non-equilibrium Stackelberg play become observationally virtually equivalent, because no observable analogue to the follower’s looking probability exists.4 Finally, the costly leader game is in some sense more well-behaved than the noisy leader game. Unlike in the noisy leader model, the equilibrium outcome and payoff correspondences of the costly leader game are both upper- and lower-hemicontinuous in the model parameter. All equilibrium outcomes and payoffs of the game with zero observation costs are therefore ‘accessible’ through equilibria of games with small but positive observation costs (lower-hemicontinuity). And all converging sequences of equilibrium outcomes or payoffs of games with positive observation costs converge to an equilibrium outcome or payoff of the zero-cost game (upper-hemicontinuity). Hence, if one is only interested in equilibrium outcomes and payoffs for very small observation costs, it suffices to study the equilibria of the zero-cost game. The remainder of this paper is structured as follows. In Section 2 we analyze an example. This example illustrates the main points of the paper. In Section 3 we develop the general model. Its pure-strategy equilibria are analyzed in Section 4 and its mixed-strategy equilibria are analyzed in Section 5. Section 6 concludes.

3 The non-moving-support assumption requires that for any action taken by the leader, the follower receives all possible signals with positive probability. 4 In the noisy leader game, equilibrium and non-equilibrium Stackelberg behavior only differ conditional on the follower receiving the unexpected ‘Cournot’ signal. But for small values of the (noise-)parameter this almost never happens. Huck and Müller (2000) encounter this problem in their experimental study of the noisy leader game. For an experimental study of the costly leader game, see Morgan and Várdy (2001).

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2. Example To focus ideas and illustrate the main points of the paper, we start out by analyzing an example. 2.1. Setup Let the normal form game g be given by L\F s c S (5, 2) (3, 1) C (6, 3) (4, 4) This is the same game as proposed by Bagwell (1995, p. 272), in the context of the noisy leader model. Here, the row player is called L and the column player is called F . These are mnemonics for ‘Leader’ and ‘Follower’, respectively. We denote L’s pure strategies in g by S and C, and F ’s pure strategies by s and c. The letters S, s and C, c stand for ‘Stackelberg’ and ‘Cournot’, respectively. The motivation is that the associated Stackelberg leader game Γ , in which L moves first and F moves second after having observed L’s action, has a unique subgame perfect Stackelberg equilibrium outcome Ss. At the same time, the simultaneousmove Cournot version, g, is dominance solvable with a unique rationalizable outcome Cc. (Dominance solvability is special to this example and is not a standard property of games analyzed in this paper.) Now, we focus attention on a modified version of the standard Stackelberg leader game Γ . This modified game is called the costly leader game and denoted by Γ ε . In Γ ε , F gets to observe L’s action before choosing s or c, if and only if he spends an amount ε. The decision whether to expend ε and observe L’s action is denoted by o = y, n. Here, we write y if F decides to observe L’s action and n if he decides not to observe L’s action. We denote F ’s pure strategy of not looking and playing s by (n, s), while the strategy of not looking and playing c is denoted by (n, c). His strategy of looking and best responding to his observation is denoted by (y, b). The probability with which F looks at L’s action is denoted by pε . Under sequential rationality, pε is equal to the probability that F plays (y, b). Player L’s pure strategies in Γ ε are the same as in g and Γ , i.e., S and C. The purestrategy sets are generalized to allow for mixing. In Γ ε , the outcome of a strategy profile is defined as the probability distribution that this profile induces on {S, C} × {s, c}. Hence, an outcome is not concerned with whether F looks at L’s action or not. We use this more restricted definition because we want to be able to compare outcomes of Γ ε with outcomes of g and Γ . Finally, the payoffs in the costly leader game Γ ε are as follows. For each outcome Ss, Cc, Sc, and Cs, the players’ payoffs in Γ ε are in principle the same as in g. For player F , however, the payoff in Γ ε also depends on whether he looks at L’s action. If F looks, the cost ε of looking is subtracted from his payoff. As solution concept we use subgame perfect equilibrium (SPE).

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2.2. Pure-strategy equilibria Let ε > 0. In the unique pure-strategy SPE of Γ ε , L always plays C and F always plays (n, c). To see why, we go through the following simple steps. First, it is easily checked that the strategy profile (C, (n, c)) is indeed a SPE. To see that it is the unique pure-strategy SPE, consider the following argument. In combination with L playing a pure strategy, ε > 0 implies that pε = 0 in equilibrium. Else, F would be paying for information he already ‘knows.’ Given that pε = 0, L must be playing his g-dominant strategy C. Then F must be playing c. Hence, (C, (n, c)) is the unique pure-strategy SPE. This means that the leader’s firstmover advantage is lost completely if we restrict attention to pure-strategy equilibria.

2.3. Mixed-strategy equilibria Next, we look at the set of mixed-strategy SPE of Γ ε when ε > 0. First note that, in any mixed-strategy equilibrium, pε must be strictly greater than zero. If pε = 0, then L will always play his g-dominant strategy C and F will always respond with c, taking us back to the unique pure-strategy equilibrium. Also, L must be mixing over S and C with strictly positive probabilities. Else, pε = 0. Finally, conditional on looking, subgame perfection forces F to play s in response to S, and c in response to C. Conditional on not looking, F may mix over s and c or play one or the other with probability 1. From these basic facts we deduce in Appendix A that Γ ε has the following mixedstrategy equilibria. • If 0 < ε
12 , there exists a SPE characterized by PrL (S) = 1 − ε, pε = 12 , PrF (y, b) = 1 1 2 and PrF (n, s) = 2 . In this equilibrium, L plays S and C with probabilities 1 − ε and ε, respectively. Player F either looks and best responds to his observation, or he does not look and plays s. Both of these actions occur with probability 12 . The outcome of this equilibrium converges to the Stackelberg outcome when ε ↓ 0. Therefore we call this equilibrium a noisy Stackelberg equilibrium. In fact, it is the unique noisy Stackelberg equilibrium of the game (see Appendix A). Note that the probability pε with which F looks at player L’s action does not depend on the cost of looking, ε. The intuition for this result is quite simple. Player F ’s mixture between looking and not looking, (pε , 1 −pε ), is such that L is indifferent between playing S and C. As neither L’s payoffs nor F ’s best replies conditional on his looking decision depend on ε, pε cannot depend on ε either. It is this prediction of constant pε that, in an experimental context, allows us to clearly distinguish between sophisticated noisy Stackelberg equilibrium play and naive, non-equilibrium Stackelberg play. By naive, non-

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equilibrium Stackelberg play we mean that the Stackelberg outcome is realized because, for low cost ε, the follower simply (almost) always looks at the leader’s action.5 • If 0 < ε
12 , there exists a SPE characterized by PrL (C) = 1 − ε, pε = 12 , PrF (y, b) = 1 1 2 and PrF (n, c) = 2 . In structure, this equilibrium is quite similar to the noisy Stackelberg equilibrium. Its outcome converges to the Cournot outcome when ε ↓ 0. Therefore we call this equilibrium a noisy Cournot equilibrium. Again, the looking probability pε does not depend on ε. In this example it is equal to the looking probability in the noisy Stackelberg equilibrium. This is not true more generally. • If ε = 12 , there exists a continuum of SPE characterized by PrL (S) = 12 , p = 12 , PrF (y, b) = 12 , and PrF (n, s) ∈ [0, 12 ]. This continuum of equilibria is a singularity. It only occurs when ε is exactly equal to 12 . • If ε > 12 , then there exists no mixed-strategy equilibrium. In that case, the pure-strategy Cournot equilibrium (C, (n, c)) is the unique SPE. When ε > 12 , looking is so expensive that there no longer exist beliefs on the leader’s probabilities of playing S and C, such that looking is a best response for the follower. It is more advantageous to play either s or c without looking. Earlier we have shown that pε > 0 in any mixed-strategy equilibrium. Hence, no mixed-strategy equilibrium exists when ε > 12 . 2.4. ε
0 For completeness, we now look at the case ε
0. When ε = 0, it is easily checked that both the Stackelberg and the Cournot outcomes are supported by SPE. (Note, however, that all the Cournot SPE involve L playing a weakly dominated strategy.) Later in the paper we show that all SPE outcomes of Γ 0 are pure. (See Proposition 5.) This result, in combination with a cell-by-cell inspection of the normal form of the costly leader game, implies that the Cournot and Stackelberg outcomes are in fact the only equilibrium outcomes of Γ 0 . From the foregoing it then follows that the equilibrium outcome and payoff correspondences as functions of ε are well behaved in this game. They are both lower and upperhemicontinuous when ε ↓ 0. That is, the limit for ε ↓ 0 of any equilibrium outcome (payoff) 5 In the noisy leader game, naive non-equilibrium Stackelberg play corresponds to the follower always

‘following the signal.’ That is, he plays s after the signal ‘S’, and c after the signal ‘C’. In the noisy Stackelberg equilibrium of this game, he only follows the signal if the signal is ‘S’. Else, he mixes. However, when the noise is small, the signal ‘C’ is very rare in equilibrium. This makes the two strategies observationally almost equivalent. See Huck and Müller (2000) and Morgan and Várdy (2001).

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of Γ ε is an equilibrium outcome (payoff) of Γ 0 . And any equilibrium outcome (payoff) of Γ 0 , including the Stackelberg outcome (payoff), is ‘accessible’ through an equilibrium σ ε of Γ ε by letting ε ↓ 0. Finally, suppose ε < 0. This corresponds to a situation in which L subsidizes F for observing his action. Subgame perfection implies that o = y. In turn, this implies that the unique SPE outcome is the Stackelberg outcome. 2.5. Equilibrium payoffs Figures 1 and 2 summarize L and F ’s equilibrium payoffs as a function of ε, in all the SPE of Γ ε . Figure 1 illustrates that, in the noisy Stackelberg equilibrium, the leader’s first-mover advantage is preserved completely. The reason is simple: Because the leader is mixing, both S and C must give him the same expected payoff. But if he plays S, the Stackelberg outcome Ss occurs with certainty. Hence, his expected equilibrium payoff is equal to his payoff in the regular Stackelberg game with perfect observability. Figure 2 illustrates that, in the noisy Stackelberg equilibrium of this example, F ’s expected payoff is 200 + ε, which is higher than his payoff in the usual Stackelberg game. Hence, an increase in the cost of looking makes the follower better off. The reason is as follows. By not always looking, player F creates the possibility of L playing C from time to time. In equilibrium, the frequency with which L plays C is increasing in ε. And, for the payoffs in this example, F is always better off when L plays C than when L plays S. Therefore, F ’s payoff increases when ε increases. However, this result does not generalize. The positive relation between F ’s observation cost and equilibrium payoff is due to the fact that F ’s Ss payoff, 2, is strictly smaller than his Cs payoff, 3. If, for example, his Cs payoff had been 1 instead of 3, F would be worse off the larger ε.

Fig. 1. Leader’s equilibrium payoffs.

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Fig. 2. Follower’s equilibrium payoffs.

In the next sections we show that most of the results derived thus far are not special features of this example but are true more generally.

3. General model We begin by adopting the base model and notation of vDH. The interested reader may want to consult vDH (1997, pp. 286–287), for details. Let g be a finite two-person game in normal form. The two players are called ‘leader’ and ‘follower’. Leader L and follower F have pure-strategy sets denoted by I and J respectively. The sets I and J are identified with the first |I |, respectively |J |, positive integers. L’s payoff when the strategy pair (i, j ) is played is uij , while F ’s payoff is vij . As in vDH, we assume that g satisfies the following regularity condition: if

(i, j ) = (k, l),

then uij = ukl

and vij = vkl .

Therefore, the players have a unique best reply to each pure strategy of their opponent. We denote F ’s best response against L’s pure strategy i ∈ I by bi ∈ J . Now, consider the sequential move version of g, in which L moves first and, after having observed L’s action, F moves second. This Stackelberg leader game Γ has a unique SPE. Player F ’s strategy in this SPE is denoted by b. That is, b prescribes bi as a response to i ∈ I . Without loss of generality, we assume that u1b1 > max uibi i=1

such that the SPE outcome of Γ is (1, b1 ). In the remainder of the paper we focus on the modified version of this game, denoted by Γ ε , in which F gets to observe L’s action before choosing j ∈ J , if and only if he

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expends an amount ε ∈ R. The decision whether to expend ε and observe L’s action is denoted by o = y, n. Here, we write y if F observes L’s action, and n if F does not observe L’s action. L’s pure-strategy set in this modified game is equal to I . F ’s pure-strategy set is AF = {y, n} × J I . Naturally, we impose the extra restriction on AF that, if F chooses o = n, he cannot condition his action on L’s action and must make the same choice for all i ∈ I. With some abuse of notation, we let (n, bi ) denote F ’s pure strategy in which he does not observe L’s move and always plays bi , while (y, b) denotes his pure strategy of observing L’s move and best responding to his observation. Strictly speaking, player F ’s strategy should also specify what he would have done if o = y (, n), even though his own strategy prescribes him to choose o = n (, y). Except for the second part of Proposition 2, these counter-factuals will not play a role. Therefore, we will ignore them in the remainder of the paper and content ourselves with describing strategies for player F which are incomplete in this strict sense. Unless specified otherwise, we set F ’s behavior in these counter-factual situations equal to behavior consistent with subgame perfection. In Γ ε , the payoff functions u(· , ·) for player L and v(· , ·) for player F are defined as follows. For a strategy profile (i, (o, j )), where i ∈ I , o ∈ {y, n} and j ∈ J , L’s payoff u(i, (o, j )) is equal to
 u i, (o, j ) = uij , while F ’s payoff is 
 vij − ε v i, (o, j ) = vij

if o = y, if o = n.

Formally, we consider the extensive form game defined by the following rules: (1) (2) (3) (4)

Player L takes an action i ∈ I . Player F chooses whether to observe i at a cost ε, i.e., he chooses o = y, n. Player F takes an action j ∈ J . Player L receives a payoff u(i, (o, j )) = uij . Player F receives




v i, (o, j ) =



vij − ε vij

if o = y, if o = n.

In the usual way, the players’ pure-strategy sets are extended to allow for mixing. A mixed strategy of L in Γ ε is denoted by r ε , while a mixed strategy of F is denoted by q ε . We write σ ε = (r ε , q ε ) for a strategy profile. Hence, r ε is a probability distribution on I ; r ε ∈ ∆(I ). And q ε is a probability distribution on AF ; q ε ∈ ∆(AF ). Obviously, also q ε must be consistent with the fact that F cannot condition his choice j ∈ J on L’s action i when he has chosen not to observe that action. We let riε denote the probability that L chooses i ∈ I , and write r ε = i if riε = 1. We use similar conventions throughout the paper. We let pε denote the probability that F gets informed, i.e., the probability that o = y. Therefore, each q ε implies a pε , which, in essence, is the marginal distribution of q ε on o = {y, n}.

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The payoff functions u(· , ·) and v(· , ·) are generalized to assign payoffs to profiles of mixed strategies σ ε = (r ε , q ε ). We denote these expected payoffs by u(r ε , q ε ) and v(r ε , q ε ), respectively. An outcome zε = z(σ ε ) of the strategy pair σ ε = (r ε , q ε ) in Γ ε is the probability distribution that σ ε induces on I × J . A pure outcome is a pair (i, j ) ∈ I × J that occurs with probability 1. We write zε = (i, j ). Note that, as defined here, an outcome of Γ ε is not a complete description of play. Specifically, it does not specify the probability pε that F looks at what L did before choosing some j ∈ J . We use this definition because we want to compare outcomes of Γ ε with outcomes of g and Γ . The appropriate equilibrium concept for this game is subgame perfect equilibrium (SPE). Unless stated otherwise, the word ‘equilibrium’ refers to an SPE.

4. Pure-strategy equilibria In this section, we restrict attention to pure-strategy SPE of Γ ε . We start with a simple lemma, which we will use in the proofs of a number of propositions. Lemma 1. Let ε > 0. In any pure-strategy SPE of Γ ε , pε = 0; i.e., the follower never looks. Proof. If L is playing a pure strategy, then F can perfectly predict L’s choice of action. In that case, looking cannot be a best response for F , because he would be spending resources (ε > 0) on learning something he is already certain of. Thus, in any pure-strategy SPE, o = n. That is, pε = 0. 2 Next, we show that the value of commitment is lost completely, if we restrict attention to pure-strategy equilibria. Proposition 2 (No value of commitment). Let ε > 0. The sets of pure-strategy Nash equilibrium (NE) outcomes of g and Γ ε coincide. Moreover, the set of pure-strategy NE outcomes of Γ ε coincides with its set of pure-strategy SPE outcomes. Proof. In this proof, psNEo stands for the set of pure-strategy NE outcomes. Similarly, psSPEo stands for the set of pure-strategy SPE outcomes. The attached subscripts indicate the game, i.e., g or Γ ε . (psNEog ⊆ psNEoΓ ε ): Assume that (i ∗ , j ∗ ) is a pure-strategy Nash equilibrium of g. Then, j ∗ = bi ∗ . If we let aF = (n, bi ∗ ), then (i ∗ , aF ) is a pure-strategy Nash equilibrium of Γ ε . This Nash equilibrium of Γ ε produces the same outcome as (i ∗ , j ∗ ) produces in g. (psNEog ⊇ psNEoΓ ε ): Now, assume that (i ∗ , aF ) is an arbitrary pure-strategy Nash equilibrium of Γ ε . Because of Lemma 1, aF = (n, bi ∗ ). In turn, this implies that in g, i ∗ is a best reply to bi ∗ . Therefore, (i ∗ , bi ∗ ) is a pure-strategy Nash equilibrium of g with the same outcome as (i ∗ , aF ) in Γ ε .

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(psNEoΓ ε ⊆ psSPEoΓ ε ): If (i ∗ , aF ) is a pure-strategy Nash equilibrium of Γ ε , then aF = (n, bi ∗ ). The only way for this Nash equilibrium not to be subgame perfect is that it prescribes non-sequentially rational behavior for player F in the unreached subgame in which o = y. In that case, we modify F ’s strategy such that he plays b conditional on the counter-factual situation o = y. Obviously, this modification makes F ’s strategy sequentially rational. Also, it does not change the outcome of the game or the payoffs to the players. To see that this modified strategy profile is still an equilibrium, note that any profitable deviation of F from this modified strategy also constitutes a profitable deviation from the original strategy. (Obviously, as moving back to the old strategy is payoff neutral, this is not profitable either.) As the modified strategy still prescribes F to always play (n, bi ∗ ), the modification cannot have created a profitable deviation for L either. Hence, we have transformed an arbitrary pure-strategy NE of Γ ε into an outcome equivalent pure-strategy SPE. (psNEoΓ ε ⊇ psSPEoΓ ε ): by definition. 2 For the sake of completeness, we now characterize the set of pure-strategy equilibria for ε
0. For ε = 0, we have: Remark 3. The set of pure-strategy SPE outcomes of Γ 0 is equal to the set of pure-strategy NE outcomes of g, plus the Stackelberg outcome. Proof. It is easily checked that all pure-strategy NE outcomes of g and the SPE outcome of Γ are indeed pure-strategy SPE outcomes of Γ 0 . Conversely, any pure-strategy SPE outcome of Γ 0 must be supported by an equilibrium strategy profile in which F either always looks at what L did, or never looks. But if he never looks, this strategy profile is an SPE of Γ 0 if and only if it is a NE of g. While, if he always looks, the profile is an SPE of Γ 0 if and only if it is the unique SPE of Γ . 2 For ε < 0 we have: Remark 4. Let ε < 0. The unique pure-strategy SPE outcome of Γ ε is equal to the Stackelberg outcome. Proof. Trivial. 2 Note that the validity of Remark 4 does not depend on the restriction to pure-strategy equilibria. Hence, the Stackelberg equilibrium is the unique equilibrium, also if we allow for mixed-strategy equilibria.

5. Mixed-strategy equilibria In order to justify a more compact notation for specific mixed strategies q ε ∈ ∆(AF ) of the follower, we note the following: If player F is sequentially rational, he always

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plays b conditional on observing L’s action. In that sense, F ’s (optimal) action conditional on looking is more or less trivial and can often be left out of a strategy description without creating much confusion. Thus, when we write q ε = (pε , t ε ), we refer to the following mixed strategy q ε of player F : Player F looks at L’s action with probability pε . Conditional on not looking he plays action j with probability tjε , j ∈ J , while, conditional on looking, he plays b. Note that F playing b conditional on looking is not reflected in the notation. When tjε = 1, we write t ε = j . Hence, F ’s ‘mixed’ strategy q ε = (0, j ) is the same as his pure strategy (n, j ), while q ε = (1, j ) is the same as (y, b). 5.1. No cost We begin by proving a property of the zero-cost game Γ 0 , that plays an important role in the proof of Proposition 7 (lower-hemicontinuity). In turn, it is the lower-hemicontinuity of the equilibrium outcome correspondence for ε ↓ 0 that implies the existence of a noisy Stackelberg equilibrium (Corollary 8). Proposition 5. All SPE outcomes z of Γ 0 are pure. That is, z = (i ∗ , bi ∗ ) for some i ∗ ∈ I . Proof. Let σ = (r, q) be an SPE of Γ 0 with outcome z. First, we check that L must be playing a pure strategy in σ . Suppose not. I.e., suppose r is such that L strictly mixes. If F has a dominant action j ∗ ∈ J against all strategies in the support of L’s mixture, then F always plays j = j ∗ and, because of the genericity of payoffs, L should not have been mixing. On the other hand, if F does not have a dominant action against the pure strategies in the support of L’s mixture, then F must always be looking at L’s action, because the cost ε of doing so is zero while the expected pay off is strictly positive. And after observing what L did, say i  , F will best respond with bi  . In that case, L should again not have been mixing, because always playing his Stackelberg action i = 1 would give him a strictly higher payoff. Contradiction. Therefore, it must be that ri ∗ = 1, for some i ∗ ∈ I . Or, equivalently, r = i ∗ . Finally, if L plays r = i ∗ in equilibrium, q must be such that F is always best responding to i ∗ . Therefore, j = bi ∗ . This completes the proof. 2 On the basis of the last proposition, one might conjecture that the set of SPE outcomes of Γ 0 only consists of the set of pure-strategy NE outcomes of g plus the Stackelberg outcome of Γ . The next proposition, however, shows that this is not true. Proposition 6. The set of SPE outcomes of Γ 0 may contain pure outcomes z = (i ∗ , bi ∗ ) that are neither Nash equilibrium outcomes of g, nor equal to the Stackelberg outcome (1, b1 ) of Γ .

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Proof. Consider the game g with the following payoff matrix: L\F S D E C

s (5, 2) (0, 0) (0, 1) (6, 3)

d (0, 0) (4, 1/2) (5, 0) (0, 0)

c (3, 1) (0, 0) (0, 0) (4, 4)

For the payoff genericity assumption to hold, slightly perturb all payoffs. The unique SPE outcome of the associate Stackelberg leader game Γ is Ss, while the unique pure-strategy NE outcome of g is Cc. These outcomes are indeed SPE outcomes of Γ 0 . However, the (pure) outcome Dd can also be supported in an SPE of Γ 0 . To see this, let L always plays D, while F plays d conditional on o = n and bestreplies  to L’s action if o = y. Moreover, F looks at L’s action with some probability p ∈ 15 , 45 .6 It is easily checked that this strategy profile is indeed an SPE of Γ 0 , for all p between 1 4 4 and 5 . This completes the proof. 2 The intuition behind this result is related to the fact that we have defined an outcome zε of Γ ε as a probability distribution on I × J , which leaves out the probability pε that F looks at L’s action. Therefore, an SPE of Γ 0 with a pure outcome z may have F mixing between looking and not looking, as long as he plays one of his actions j ∈ J with probability one. We have used this fact to construct a game Γ 0 with a mixed-strategy SPE entailing a pure outcome (i ∗ , bi ∗ ) that is neither a NE outcome of g, nor equal to the Stackelberg outcome of Γ . In this equilibrium, F looks sufficiently often to make it unprofitable for L to switch to his Cournot action C, while, at the same time, F looks sufficiently rarely as to prevent L from profitably switching to his Stackelberg action S. 5.2. Lower-hemicontinuity and the existence of noisy Stackelberg equilibria We now show that there always exists a SPE of Γ ε , whose outcome converges to the Stackelberg outcome when ε ↓ 0. Such an equilibrium is called a noisy Stackelberg equilibrium. We derive this result by proving that the equilibrium outcome correspondence of Γ ε is lower-hemicontinuous for ε ↓ 0. That is, we show that all SPE outcomes of Γ 0 are ‘accessible’ through SPE outcomes of Γ ε when ε ↓ 0. Note that lower-hemicontinuity of the equilibrium outcome correspondence is a weaker statement than lower-hemicontinuity of the equilibrium strategy correspondence. This is because, in our model, an outcome of Γ ε is not concerned with the probability pε of looking, but only with the probabilities of the actions i ∈ I and j ∈ J . In fact, the game analyzed in Section 2 constitutes a counter-example to lowerhemicontinuity of the equilibrium strategy correspondence. There, the strategy profile (S, (y, b)) is a SPE of Γ 0 , with outcome Ss. And though there exists an equilibrium σ ε of Γ ε whose outcome converges to Ss when ε ↓ 0—namely, the noisy Stackelberg equilibrium—there is no equilibrium of Γ ε in which the probability of looking goes to 1. 6 Of course, perturbation of the payoffs changes these boundaries slightly.

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Proposition 7 (Lower-hemicontinuity). Suppose z is the outcome of a SPE of Γ 0 . Then there exists an SPE σ ε = (r ε , q ε ) of Γ ε , with an outcome zε , such that zε → z when ε ↓ 0. The proof is quite straightforward but long. First, we take an arbitrary SPE of Γ 0 , which by Proposition 5 we know to have a pure outcome z = (i ∗ , bi ∗ ). Then we construct an equilibrium of Γ ε , whose outcome converges to z when ε ↓ 0. The full proof can be found in Appendix B.1. Note that lower-hemicontinuity of the equilibrium outcome correspondence implies lower-hemicontinuity of the equilibrium payoff correspondence. The existence of a noisy Stackelberg equilibrium is a direct consequence of Proposition 7. Corollary 8. All noisy leader games Γ ε have an SPE σ ε whose outcome converges to the Stackelberg outcome when ε ↓ 0. Such an equilibrium is called a noisy Stackelberg equilibrium. Noisy Cournot equilibria. If there exists a pure-strategy Cournot equilibrium (i ∗ , j ∗ ) of g, then, by Proposition 7, there also exists a noisy Cournot equilibrium of Γ ε , whose outcome converges to the Cournot outcome when ε ↓ 0. Note, however, that this is a trivial statement, because the pure-strategy profile (i ∗ , (n, j ∗ )) qualifies as a ‘noisy’ Cournot equilibrium of Γ ε . In fact, it may or may not be the only noisy Cournot equilibrium associated with (i ∗ , j ∗ ). Existence of a pure-strategy Cournot equilibrium is not guaranteed. For example, the simultaneous-move game L\F d e D (2, 1) (1, 2) E (0, 3) (3, 0) which is essentially ‘matching pennies’ with generic payoffs, has none. It does have a mixed-strategy equilibrium with probability mixtures ( 34 , 14 ) for L and ( 12 , 12 ) for F . The outcome distribution induced by this equilibrium is not accessible through equilibria of the sequential move costly leader game when ε ↓ 0. This follows from Proposition 5. Note that in the example above, and others like it, there is a first-mover disadvantage. While L makes 32 in the simultaneous-move game g, he only makes 1 in the Stackelberg game Γ . 5.3. Some properties of noisy Stackelberg equilibria In this subsection we show that, for sufficiently small ε, the probability that the follower looks at the leader’s action is independent of the cost of looking. Moreover, this probability is the same for all noisy Stackelberg equilibria. We also show that the leader’s value of commitment is preserved completely in all noisy Stackelberg equilibria of Γ ε . This is a strengthening of the standard vDH finding for the noisy leader game. For the noisy leader game, the leader’s value of commitment is only preserved asymptotically. Finally, we prove

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uniqueness of the noisy Stackelberg equilibrium in costly leader games based on 2 × |J | games g. In the following propositions, Γ ε is an arbitrary costly leader game with generic payoffs and ε > 0. Full proofs can be found in Appendix B.2. Proposition 9 (Independence from ε). There exists an ε > 0 such that for all ε < ε¯ the follower’s probability of looking in all noisy Stackelberg equilibria σ ε of Γ ε is independent of ε and equal to some p∗ ∈ [0, 1]. Hence, the probability of looking is independent of the cost of looking and the same for all noisy Stackelberg equilibria. The intuition behind the independence of the follower’s probability of looking on the cost of looking is as follows. In a noisy Stackelberg equilibrium σ ε , the value of pε is chosen to keep the leader indifferent between his Stackelberg action i = 1 and at least one other action. However, neither L’s payoff nor F ’s best response conditional on his looking decision depend on ε. Therefore, the equilibrium probability pε that F looks cannot depend on ε either. Proposition 10 (Stackelberg payoff preservation). There exists an ε > 0 such that for all ε < ε the leader’s payoff in all noisy Stackelberg equilibria σ ε of Γ ε is equal to his payoff in the conventional Stackelberg leader game Γ . Hence, the leader’s value of commitment is preserved completely. The intuition behind the full preservation of the leader’s first-mover advantage in all noisy Stackelberg equilibria is as follows. As the leader is mixing between his Stackelberg action i = 1 and at least one other action i  = 1, both 1 and i  must give him the same expected payoff. If he plays 1, the Stackelberg outcome (1, b1 ) occurs with certainty. Hence, his expected equilibrium payoff in the noisy Stackelberg equilibrium must be equal to his payoff in the regular Stackelberg game with perfect observability. Proposition 11 (Uniqueness). If g is 2 × |J |, |J |  1, then there exists an ε > 0, such that for all ε < ε the noisy Stackelberg equilibrium σ ε of Γ ε is unique. When |I | > 2, however, the noisy Stackelberg equilibrium may not be unique. For example, the costly leader game based on L\F s c S (5, 2) (2, 1) E (7, 1) (3, 2) C (6, 3) (4, 4) has (infinitely) many noisy Stackelberg equilibria, in which the leader either mixes between S and C, between S and E, or between all three strategies. However, the follower’s looking probability is always the same.

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5.4. Upper-hemicontinuity Finally, we prove upper-hemicontinuity of the equilibrium strategy correspondence of Γ ε when ε ↓ 0. That is, we show that the limit for ε ↓ 0 of any SPE σ ε of Γ ε is an SPE of Γ 0 . In combination with the previously proved lower-hemicontinuity of the equilibrium payoff and outcome correspondences, this implies that the equilibrium payoffs and outcomes of Γ 0 are good approximations of those of Γ ε when ε is small. Hence, as long as observation costs are small and one is only interested in payoffs and outcomes, it suffices to study the equilibria of the zero-cost game. Proposition 12 (Upper-hemicontinuity). Let σ ε be an SPE of Γ ε . If σ = limε↓0 σ ε , then σ is an SPE of Γ 0 . This result is proved in two steps. First we prove that any converging equilibrium sequence of Γ ε induces an outcome sequence that converges to a pure outcome. Given Proposition 5, this is a necessary condition, albeit not sufficient. Then we prove that the limit of any converging equilibrium sequence of Γ ε whose outcomes converge to a pure outcome must be an equilibrium of Γ 0 . The full proof can be found in Appendix B.3.

6. Conclusion In this paper we have studied how the presence of observation costs on the part of the follower affects a leader’s first-mover advantage in a broad class of leader-follower games. In these costly leader games, the follower gets to observe the leader’s action if and only if he expends an amount ε. We have shown that the leader’s value of commitment is lost completely in all pure-strategy equilibria of the costly leader game. However, if the observation cost is not too large, there always exists a mixed-strategy equilibrium that completely preserves the leader’s value of commitment. In such a noisy Stackelberg equilibrium, the probability that the follower looks at leader’s action is independent of the cost of looking. Evolutionary arguments suggest that noisy Stackelberg equilibria will not survive. Selten (1980) shows that, in asymmetric games like the costly leader game, equilibria are evolutionary stable if and only if they are strict. Hofbauer and Sigmund (1988) prove an analogous result for asymptotic stability under the standard replicator dynamic. For the costly leader game this implies that only the pure-strategy Cournot equilibria are stable. Apart from asymptotic and evolutionary stability, these pure-strategy Cournot equilibria also have the desirable property of persistence (Kalai and Samet, 1984), and they belong to a primitive formation (Harsanyi and Selten, 1988). Hence, standard equilibrium-selection arguments suggest that arbitrarily small observation costs on the part of the follower may indeed lead to a complete loss of the leader’s value of commitment. In fact, evolutionary analysis of the noisy leader game leads to an analogous conclusion. (See Oechssler and Schlag, 2000.)

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On the basis of the cost-independence property of the follower’s looking behavior in noisy Stackelberg equilibria, we have argued that the costly leader game is particularly suited for experimental testing of the robustness of the first-mover advantage. Unlike the noisy leader game, it allows for a clear cut test of the behavioral relevance of both the Bagwell (1995) hypothesis that the value of commitment is lost completely, and the Van Damme and Hurkens (1997) hypothesis that it will survive through coordination on a noisy Stackelberg equilibrium. In a companion paper to this one (Morgan and Várdy, 2001), we report on the results of such an experiment. Finally, a natural next step in this line of research is the integration of the noisy and costly leader models. In such a world, the follower has the option of buying an imperfect signal about the leader’s action. It is clear that the loss of commitment value in all pure-strategy equilibria carries over to this environment. Also, in the context of the example discussed in Section 2, a noisy Stackelberg equilibrium can be derived quite easily. Establishing the existence of noisy Stackelberg equilibria more generally, and studying their properties, remain tasks for future research. Acknowledgments The author thanks Avinash Dixit, John Morgan, two anonymous referees, and especially Hugo Sonnenschein, for their comments and suggestions.

Appendix A. Derivation of mixed-strategy equilibria In this appendix we derive the set of mixed-strategy SPE of the example studied in Section 2. Let ε > 0. First, recall that in any mixed-strategy equilibrium pε must be strictly greater than zero. Also, conditional on looking, subgame perfection forces F to play s in response to S, and c in response to C. Conditional on not looking, F may mix over s and c or play one or the other with probability 1. We now systematically go over all possible types of mixed-strategy equilibria. First we investigate whether and when Γ ε allows for almost-completely-mixed-strategy equilibria. That is, equilibria in which L mixes over S and C, while F mixes over looking and not looking and, conditional on o = n, also over s and c. If o = y, however, subgame perfection forces F to play his pure best response b. A.1. Almost-completely-mixed-strategy equilibria In an equilibrium in which F strictly mixes over s and c conditional on not looking, he must be indifferent between the two actions. In the following indifference condition, rS denotes PrL (S), i.e., the probability that L plays S. If F mixes over s and c conditional on o = n, it must be that 2rS + 3(1 − rS ) = rS + 4(1 − rS ). Hence, rS = 12 , with a conditional payoff for F equal to 52 .

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The fact that pε > 0 implies that F is also indifferent between looking and not looking. Therefore,     1 5 1 +4 1− = . −ε + 2 2 2 2 Solving for ε, we get 1 ε= . 2 Denote by ts the probability Pr(s | n) that F plays s conditional on not looking. Then, L’s indifference condition for playing S and C is
 
  



 5 pε + 1 − pε ts + 3 1 − pε (1 − ts ) = 6 1 − pε ts + 4 pε + 1 − pε (1 − ts ) . In this equation, ts cancels out. We get 1 pε = . 2 We conclude that, only in the special case where ε = 12 , there exists a continuum of almost-completely-mixed equilibria, characterized by rS = 12 , pε = 12 and ts ∈ [0, 1]. A.2. Partly-mixed-strategy equilibria Now we proceed to the set of partly-mixed equilibria of this game. Recall that, in any mixed-strategy equilibrium, F must mix over looking and not looking, because pε has to be > 0. If at the same time ε = 12 , then player F cannot be indifferent between playing s and c conditional on not looking. Therefore it must be that ts ∈ {0, 1} in equilibrium. That is, conditional on not looking, F either plays the pure action s, corresponding to ts = 1, or plays the pure action c, corresponding to ts = 0. Postulate that ts = 1 in equilibrium. This implies that rS must be  12 . Else, ts = 1 is not a best response for the follower. The indifference condition for F mixing over looking and not looking is −ε + 2rS + 4(1 − rS ) = 2rS + 3(1 − rS ) which reduces to rS = 1 − ε. Among other things, this implies that ε must be
12 . Else, rS would not be  12 . Player L’s indifference condition for mixing over S and C is 

5pε + 5 1 − pε = 6 1 − pε + 4pε implying that 1 pε = . 2 Hence, for all ε ∈ (0, 12 ), there exists an SPE σ ε characterized by rS = 1 − ε, pε = 12 , and ts = 1.

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Next, postulate that ts = 0 in equilibrium. That is, F always plays c conditional on not looking. This implies that rS must be
12 . The indifference condition for F mixing over o = y, n is −ε + 2rS + 4(1 − rS ) = rS + 4(1 − rS ) giving rS = ε. Again, this implies that ε must be
12 for such an equilibrium to exist. Player L’s indifference condition for mixing over S and C is  

5pε + 3 1 − pε = 4pε + 4 1 − pε . Hence, 1 pε = . 2 Therefore, for all ε ∈ (0, 12 ), there also exists an equilibrium σ ε characterized by rS = ε, = 12 , and ts = 0. As we have gone over all the possible types of mixed and partly mixed SPE, we have given a characterization of all mixed-strategy SPE of this game. Therefore, the noisy Stackelberg equilibrium derived earlier is indeed the unique noisy Stackelberg equilibrium. From the foregoing it also follows that, for all ε > 12 , the only SPE is the pure-strategy SPE with outcome zε = Cc. pε

Appendix B. Proofs B.1. Lower-hemicontinuity In this subsection we prove lower-hemicontinuity of the equilibrium outcome correspondence. First we introduce the following definitions. Definition 13. Define ζi ∈ [0, 1), to be the probability such that the following statement is true: If F believes that riε  ζi , then, conditional on not looking, j = bi is a best reply for F against r. Genericity of payoffs makes that each bi is a strict best reply to a pure action i ∈ I of the leader. Therefore, the threshold value ζi exists, is unique, and is strictly smaller than 1. Note that ζi does not depend on ε. Definition 14. For all i  , i  ∈ I such that bi  = bi  , define δiε i  to be such that δiε i  (vi  bi  − vi  bi  ) = ε.

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If bi  = bi  , then vi  bi  < vi  bi  . For sufficiently small ε, the threshold value δiε i  therefore exists and is unique. Also, δiε i  ↓ 0 when ε ↓ 0. Note that the definition of δiε i  implies the following: If L mixes over i  and i  with probabilities ri  = 1 − δiε i  , ri  = δiε i  , and ε is so small that 1 − δiε i  > ζi  , then F is just indifferent between playing (n, bi  )and (y, b). Now we are ready to prove lower-hemicontinuity of the equilibrium outcome correspondence. Proof of Proposition 12 (Lower-hemicontinuity). From Proposition 5 we know that z is pure, i.e., z = (i ∗ , bi ∗ ) for some i ∗ ∈ I . The SPE σ of Γ 0 that supports z must have F play bi ∗ if o = n, and b if o = y. We now prove the proposition by constructing an equilibrium σ ε = (r ε , q ε ) of Γ ε , whose outcome converges to z when ε ↓ 0. We define P to be  ε P = p ∈ [0, 1]: i ∗ ∈ BRΓL (p, bi ∗ ) . In words: The set P consists of all looking probabilities p ∈ [0, 1], such that action i ∗ is a best reply for L against F playing the strategy (p, bi ∗ ) in Γ ε . The set P has the following properties: • P is closed. • P does not depend on ε. • P is non-empty. Figure 3 illustrates these points. It graphs L’s payoffs associated with his various pure strategies as a function of p, when F plays (p, bi ∗ ). Closedness follows from the fact that best replies are not ‘strict.’ P ’s independence of ε follows from the fact that neither L’s payoffs uij , i ∈ I , j ∈ J , nor the strategy (p, bi ∗ ) depend on ε. Non-emptiness then follows from the following argument. Note that i ∗ is part of the SPE outcome z = (i ∗ , bi ∗ ) of Γ 0 . Furthermore, it must be true that player F ’s strategy in the equilibrium that supports z = (i ∗ , bi ∗ ) is given by (p, ˘ bi ∗ ) for some p˘ ∈ [0, 1]. Finally, because P does not depend on ε, i ∗ must also be a best reply to (p, ˘ bi ∗ ) in Γ ε . Now, set F ’s strategy equal to q ε = (p∗ , bi ∗ ), where p∗ = min P . Note that p∗ —and thereby q ε —do not depend on ε, because P does not depend on ε. If p∗ = 0, then i ∗ must be a best reply to bi ∗ , which makes (i ∗ , bi ∗ ) a NE of g. Therefore, σ ε = (i ∗ , (n, bi ∗ )) is an SPE of Γ ε with outcome (i ∗ , bi ∗ ) and we are done. Hence, from here on we assume that p∗ > 0. When p∗ > 0, then L must have (at least) two pure best responses against (p∗ , bi ∗ ).  That is, i ∗ and some i = i ∗ .

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Fig. 3. L’s payoffs associated with his various pure actions i, when F plays (p, bi ∗ ).

Moreover, because p∗ > 0 by assumption, it cannot be that player F ’s best response to i ∗ is the same as his best response to i  , i.e., it must be that bi ∗ = bi  . To see why, suppose that bi ∗ = bi  . Under the strategy (p∗ , bi ∗ ), p∗ > 0, F plays bi ∗ conditional on not looking and, conditional on looking, best-responds to his observation. However, if F ’s best response to observing i  is the same as his best response to observing i ∗ , then it does not matter for player L whether F looks or not. In that case, L must also be indifferent between playing i ∗ and i  against (p, bi ∗ ) if p = 0. But this would imply that p∗ > 0 is not equal to min P . As p∗ is in fact defined as min P , we have a contradiction. Therefore, bi ∗ = bi  . Now, let L only mix over i ∗ and i  , and let r ε be equal to this mixture. Our goal is to turn the strategy profile (r ε , (p∗ , bi ∗ )) into a SPE of Γ ε whose outcome converges to (i ∗ , bi ∗ ) when ε ↓ 0. To make F willing to choose p = p∗ > 0 under the constraint that he has to play some strategy of the type (p, bi ∗ ), let L play i ∗ and i  with probabilities riε = δiε∗ i  , and riε∗ = 1 − δiε∗ i  . (Note that the probability δiε∗ i  is well-defined because bi ∗ = bi  .) In that case, the extra payoff to player F from being able to respond to i  by bi  instead of bi ∗ exactly compensates for the cost ε of looking. This makes p = p∗ constrained incentive compatible for F . Moreover, if F indeed plays (p∗ , bi ∗ ), then mixing over i ∗ and i  is incentive compatible for L. Finally, to make F ’s strategy (p∗ , bi ∗ ) unconstrained optimal, we have to make sure that it is indeed incentive compatible for F to choose bi ∗ conditional on being uninformed about L’s action. If ε is chosen so small that riε∗ = 1 − δiε∗ i   ζi ∗ then L plays i  sufficiently rarely, such that playing bi ∗ is indeed the optimal choice for F .

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Thus, for sufficiently small ε, the strategy profile  



 σ ε = riε , riε∗ , q ε = δiε∗ i  , 1 − δiε∗ i  , (p∗ , bi ∗ ) is an SPE of Γ ε . Finally, because limε↓0 δiε∗ i  = 0, we have that riε∗ → 1 when ε ↓ 0. Therefore, zε → (i ∗ , bi ∗ ). This completes the proof. 2 B.2. Properties of noisy Stackelberg equilibria In this subsection we prove Propositions 9–11. The following lemma is used in the proofs of all three propositions. Lemma 15. Suppose σ ε = (r ε , q ε ) is a noisy Stackelberg equilibrium of Γ ε . Then there exists an ε > 0 such that for all ε < ε the following statements are true. • Player F ’s equilibrium strategy is q ε = (pε , b1 ), pε ∈ [0, 1]. • If the Stackelberg outcome z = (1, b1 ) constitutes a NE of g, then σ ε = (1, (n, b1 )) is the unique ‘noisy’ Stackelberg equilibrium of Γ ε . • If the Stackelberg outcome z = (1, b1 ) does not constitute a NE of g, then q ε = (pε , b1 ) with 0 < pε < 1. Proof. We prove the claims one by one. The first claim is that q ε = (pε , b1 ) for sufficiently small ε. Because σ ε = (r ε , q ε ) is a noisy Stackelberg equilibrium, r1ε → 1 when ε ↓ 0. This implies that there exists an ε > 0 such that, for all ε < ε, r1ε > ζ1 . By definition of ζ1 , q ε must then be such that F always plays b1 conditional on not looking. Conditional on looking, subgame perfection implies that F plays b. But then q ε = (pε , b1 ). This completes the proof of the first part of the lemma. The second part of the lemma consists of the following claim. If z = (1, b1 ) is a NE of g, then, for sufficiently small ε, q ε = (n, b1 ) and σ ε = (1, (n, b1 )) is the unique ‘noisy’ Stackelberg equilibrium of Γ ε . From the first part of the lemma we know that F plays q ε = (pε , b1 ), pε ∈ [0, 1], when ε is sufficiently small. If z = (1, b1) is a NE of g, then i = 1 is L’s unique best reply to F playing q ε = (pε , b1 ) in Γ ε , for all possible pε ∈ [0, 1]. Why? Conditional on F looking, i = 1 must be the unique best reply to q ε = (pε , b1 ) because z = (1, b1 ) is the Stackelberg outcome. Conditional on not looking, i = 1 must be the unique best reply to q ε = (pε , b1 ) because z = (1, b1 ) is a NE of g. Therefore, i = 1 is also the unique unconditional best reply to q ε = (pε , b1 ), for all possible pε ∈ [0, 1]. Now, if L always plays the pure action i = 1, then pε = 0 by lemma 1. In turn, this implies that σ ε = (1, (n, b1)) is the unique ‘noisy’ Stackelberg equilibrium of Γ ε , for ε < ε¯ . This completes the proof of the second part of the lemma.

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The last part of the lemma claims that there exists an ε > 0 such that for all ε < ε, 0 < pε < 1 whenever z = (1, b1) is not a NE of g. To see why this is true suppose that, for some ε, pε = 1. Then it must be that r1ε = 1, which means that L is playing a pure strategy. Therefore, pε = 0 by Lemma 1. Contradiction. Alternatively, suppose pε = 0. Because (1, b1 ) is not a NE of g, it must be that r1ε = 0. If this were to happen for some ε < ε¯ no matter how small we choose ε¯ , then the outcome zε of σ ε could not converge to (1, b1 ). Contradiction. This completes the proof of the last part of the lemma. 2 With this lemma in hand we first prove independence of pε from ε in all noisy Stackelberg equilibria of Γ ε . Proof of Proposition 9 (Independence from ε). Suppose σ ε = (r ε , q ε ) is a noisy Stackelberg equilibrium of Γ ε . By definition, this implies that its outcome converges to (1, b1 ) when ε ↓ 0. If z = (1, b1) is a NE of g, and ε is sufficiently small, then we know from the second part of Lemma 15 that pε = 0 and σ ε = (1, (n, b1 )). Hence, if z = (1, b1 ) is a NE of g then, for sufficiently small ε, the follower’s probability of looking in all noisy Stackelberg equilibria is independent of ε and equal to p∗ = 0. Moreover, the leader’s payoff is equal to his Stackelberg payoff (which, in this case, is the same as the NE payoff). Thus we are done. Alternatively, suppose that the Stackelberg outcome z = (1, b1) is not a NE of g. From the first part of Lemma 15 we know that q ε = (pε , b1 ) for ε sufficiently small. Moreover, by the last part of the same lemma we know that 0 < pε < 1. Now, by Lemma 1, pε > 0 implies that L cannot be playing a pure strategy but, instead, must be strictly mixing over i = 1 and at least one other strategy. Figure 4 graphs L’s payoffs associated with his various pure strategies as a function of p when F plays (p, b1 ). At p = 1, all lines must be strictly below the horizontal (i = 1)-

Fig. 4. L’s payoffs associated with his various pure actions i, when F plays (p, b1 ).

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line, because i = 1 is L’s unique best reply to F playing the pure strategy (y, b). At p = 0, there must be at least one line, say the (i = i  )-line, that lies strictly above the (i = 1)-line. The reason is that z = (1, b1 ) is not a NE of g and, therefore, i = 1 is not a best reply to F playing the pure strategy (n, b1 ). Also, all lines must be straight lines, because of the linearity of L’s payoffs in p. Together, these observations imply that all (i = i  , i  , . . .)-lines starting from above the (i = 1)-line at p = 0 intersect the (i = 1)-line exactly once at some 0 < p < 1. Moreover, all lines starting from below the (i = 1)-line at p = 0 must remain below it for all p
1. (By payoff genericity, there is no line that starts exactly at the same point as the (i = 1)line.) For each i = i  , i  , . . . , the unique value of p at which the intersection occurs is denoted by pi  , pi  , . . . , respectively. We denote the set of pi  , pi  , . . . by P  , while max P  is denoted by p∗ . Now, suppose that for some ε the equilibrium looking probability pε in q ε = (pε , b1 ) is strictly greater than p∗ . In that case, i = 1 is L’s unique best reply to F playing q ε = (pε , b1 ) and L cannot be mixing. Contradiction. If, for some ε, the equilibrium looking probability pε in q ε = (pε , b1 ) is strictly smaller than p∗ , then i = 1 is not a best reply for L against F playing q ε = (pε , b1 ). In that case, r1ε = 0. For sufficiently small ε, however, this cannot be true because r1ε → 1 by assumption. Therefore, for all ε smaller than some positive threshold ε¯ , pε must be equal to p∗ at all times. As P  does not depend on ε, p∗ does not depend on ε either. Hence there exists an ε¯ > 0 such that for all ε < ε¯ the follower’s probability of looking in all noisy Stackelberg equilibria is independent of ε and equal to p∗ = max P  . This concludes the proof. 2 Next, we prove the full preservation of the leader’s first-mover advantage in all noisy Stackelberg equilibria of all costly leader games Γ ε . Proof of Proposition 10 (Stackelberg payoff preservation). By definition, for sufficiently small ε, the leader plays the action i = 1 with strictly positive probability in any noisy Stackelberg equilibrium. Therefore, his expected noisy Stackelberg equilibrium payoff must be equal to the payoff he gets from playing i = 1. From the first part of Lemma 15 we know that q ε = (pε , b1 ) for ε sufficiently small. The strategy profile (1, (pε , b1 )) always results in the Stackelberg outcome z = (1, b1 ), irrespective of the value of pε . Hence, the leader’s payoff from playing i = 1 is equal to his Stackelberg payoff u1b1 . This implies that his expected noisy Stackelberg equilibrium payoff is equal to u1b1 as well. This concludes the proof. 2 Finally, we prove uniqueness of the noisy Stackelberg equilibrium in all Γ ε based on 2 × |J |, |J |  1, normal form games g. Proof of Proposition 11 (Uniqueness). Let Γ ε be a costly leader game based on a 2 × |J | game g. From Proposition 7 we know that there is an ε¯ such that for all ε < ε¯ a noisy Stackelberg equilibrium σ ε of Γ ε exists. Now, let σ ε = (r ε , q ε ) be such a noisy Stackelberg equilibrium of Γ ε .

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If z = (1, b1) is a NE of g, and ε is sufficiently small, then we know from the second part of Lemma 15 that σ ε = (1, (n, b1 )). Therefore, there exists an ε¯ > 0 such that for all ε < ε¯ any noisy Stackelberg equilibrium σ ε of Γ ε must be equal to (1, (n, b1 )). Hence, we are done. Alternatively, suppose that the Stackelberg outcome z = (1, b1) is not a NE of g. From the first part of Lemma 15 we know that q ε = (pε , b1 ) for ε sufficiently small. Moreover, by the last part of the same Lemma we know that under the same conditions 0 < pε < 1. Now, pε > 0 implies by Lemma 1 that L cannot be playing a pure strategy but, instead, must be strictly mixing in equilibrium. By assumption, L has only two actions i = 1, 2. Hence, he must be playing both of them with strictly positive probabilities. Because F is ε playing (pε , b1 ) and 0 < pε < 1, these probabilities must be equal to r1ε = 1 − δ12 and ε ε ε r2 = δ12 , respectively. And because L is mixing over i = 1, 2, p must be such that 



u 1, pε , b1 = u 2, pε , b1 . Because z = (1, b1 ) is not a NE of g, 

u 1, (n, b1) = u1b1 < u2b1 = u 2, (n, b1 ) . Because z = (1, b1 ) is the Stackelberg outcome of Γ ,

 u 1, (y, b) = u1b1 > u2b2 = u 2, (y, b) . Then, by linearity of u(1, (pε , b1 )) and u(2, (pε , b1 )) in pε , there exists a unique pε such that u(1, (pε , b1 )) = u(2, (pε , b1 )). We denote this solution by p2ε .7 This means that, for sufficiently small ε, any noisy Stackelberg equilibrium σ ε must be ε , δ ε ). equal to (r ε , (p2ε , b1 )), where r ε = (r1ε , r2ε ) = (1 − δ12 12 This completes the proof. 2 B.3. Upper-hemicontinuity In this section we prove upper-hemicontinuity of the equilibrium strategy correspondence. We begin by proving the following lemma. Lemma 16. Let σ ε = (r ε , q ε ) be an SPE of Γ ε . If σ ε converges when ε ↓ 0, then its outcome zε converges to a pure outcome z = (i ∗ , bi ∗ ), for some i ∗ ∈ I . Proof. If |I | = 1 then the result is trivial. Therefore, assume that |I | > 1. We begin by proving that, for any equilibrium σ ε = ε (r , q ε ) of Γ ε that converges when ε ↓ 0, there exists an i ∗ ∈ I such that riε∗ → 1 when ε ↓ 0. Suppose not. That is, suppose there is no i ∗ ∈ I such that riε∗ → 1 when ε ↓ 0. Since r ε is convergent by hypothesis, all coordinates must be bounded strictly away from 1. As |I | 7 It can easily be checked that p ε does not depend on ε. A formal argument is given in the proof of 2 Proposition 9.

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is finite and the probability mass has to go somewhere, r ε has (at least two) components, i  , i  , . . . , that are also bounded away from 0. Denote this set of actions i  , i  , . . . by I  . / I  , riε → 0 when ε ↓ 0. This last fact implies the following: By definition of I  , for all i ∈ Suppose F has a dominant action j ∗ in g against all i ∈ I  . In Γ ε it must then be true that, conditional on o = n, F always plays j ∗ when ε is sufficiently small. Dominance of j ∗ against all i ∈ I  further implies that, conditional on o = y and observing some i  ∈ I  , F again plays j ∗ . This, however, means that L has a profitable deviation from his postulated behavior, irrespective of the value of pε . Whenever he previously played some action i  in I  —all of which are played with strictly positive probabilities bounded away from 0—he should switch to the unique constrained best reply in I  against j ∗ . If F does not have a dominant action against all i ∈ I  , there are (at least) two actions i  , i  ∈ I  such that bi  = bi  . That is, F ’s best reply to L playing action i  is different from F ’s best reply to i  . Therefore, F derives a strictly positive value from observing what L did. Moreover, this value is bounded away from 0 when ε ↓ 0 by the fact that riε , i ∈ I  , is bounded away from 0. For sufficiently small ε, F will therefore always choose to observe L’s action and play his (unique) best reply to what he observes L has played. In that case, however, L should not have been mixing. Instead, he should have played i = 1 all the time. Therefore, both cases lead to a contradiction. This implies that, whenever σ ε = (r ε , q ε ) converges, there exists an i ∗ ∈ I such that riε∗ → 1 when ε ↓ 0. If riε∗ → 1 then, for sufficiently small ε, riε∗ > ζi ∗ . By definition of ζi ∗ , this makes F play bi ∗ conditional on not looking. Conditional on looking he best-replies to his observation. Because riε∗ → 1, the probability that this best reply consists of playing bi ∗ converges to 1 as well. Therefore, when ε ↓ 0, zε → (i ∗ , bi ∗ ) for some i ∗ ∈ I . This completes the proof. 2 Now, we use this lemma to prove upper-hemicontinuity of the equilibrium strategy correspondence in ε = 0. Proof of Proposition 12 (Upper-hemicontinuity). From Lemma 16 we know that, if an equilibrium strategy profile σ ε = (r ε , q ε ) of Γ ε converges when ε ↓ 0, its outcome converges to (i ∗ , bi ∗ ) for some i ∗ ∈ I . Therefore, for sufficiently small ε, riε∗ > ζi ∗ . By definition of ζi ∗ , it must be that q ε = (pε , bi ∗ ). (Recall that q ε = (pε , bi ∗ ) means that F looks with probability pε . Conditional on not looking F plays bi ∗ and conditional on looking F plays b.) Any converging equilibrium strategy profile σ ε of Γ ε therefore converges to a profile σ = (r, q) in which L plays some pure action i ∗ ∈ I and F plays q = (p, bi ∗ ), where p ∈ [0, 1]. Because ε = 0 in the limit, irrespective of the limit value of p, the limit strategy q is a best reply to L playing i ∗ . Therefore, the only way for the limit profile σ not to be an equilibrium of Γ 0 is if L ’s action i ∗ is not a best reply to the limit strategy q of F .

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Now, suppose that in the limit some i ∗∗ does strictly better against q than i ∗ . By continuity of L’s payoffs and convergence of q ε to q when ε ↓ 0, it must then be that there exists an ε-neighborhood around 0 where i ∗∗ does strictly better against q ε than i ∗ . However, from Lemma 16 we know that riε∗ → 1. Contradiction. Therefore, the profile σ must be an equilibrium of Γ 0 . This completes the proof. 2

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