Dragon Slaying with Ambiguity: Theory and Experiments

This paper studies the impact of ambiguity in the best shot and weakest link models of public good provision. The models are first analysed theoretically. Then we conduct experiments to study how ambiguity affects behaviour in these games. We test whether subjects' perception of ambiguity differs between a local opponent and a foreign one. We find that an ambiguity safe strategy, is often chosen by subjects. This is compatible with the hypothesis that ambiguity aversion influences behaviour in games. Subjects tend to choose contributions above (resp. below) the Nash equilibrium in the Best Shot (resp. Weakest Link) model.


Introduction
This paper reports some theoretical results on how ambiguity a¤ects behaviour in the best shot and weakest link models of public good provision. We then proceed to study experimentally the impact of ambiguity in these models.
Public goods are goods which can be consumed by everybody. We study situations where individuals make voluntary contributions to the provision of the public good. Due to the collective nature of the good everybody enjoys the same amount of it, irrespective of their own contribution. The usual assumption is that the amount of the public good available is a function of the sum of all individual contributions. Ambiguity in the standard public goods model has been previously studied in Eichberger and Kelsey (2002) and Bailey, Eichberger, and Kelsey (2003). These models have been tested experimentally by Di Mauro and Castro (2011).
An alternative, is the best-shot model, where production of the public good is determined by the maximum contribution made by an individual in the community. This may be represented as: u i (x i ; x i ) = max fx 1 ; :::; x n g cx i ; where x i denotes the contribution of individual i and c denotes the marginal cost of a contribution. In this case, making a large contribution, x i results in a large cost, cx i , but the bene…t accrues to all members of the group. This model may be illustrated by a medieval village that is besieged by a dragon. It is only the knight endeavouring to slay the dragon, who bears the cost -in this case, the chance that he will be burnt to a crisp by the dragon. However, once the dragon is slain, the bene…ts of a dragon-free village are enjoyed equally by all the village folk! A "whistle-blower" who exposes corruption may be seen as a modern-day dragon-slayer. He bears the burden that comes with the act of whistle-blowing, though the bene…t of his act accrues to the general public.
A third possibility is the weakest link model, in which provision of the public good is a function of the minimum of the individual contributions. It may be represented as: u i (x i ; x i ) = min fx 1 ; :::; x n g cx i . It may be noted that making a large contribution, x i ; would have a large cost, cx i ; but does not guarantee a large payo¤, since the minimum contribution made within the group of individuals would determine the level of the public good.
This model can be illustrated by the example of a small island community that must build sea defences to protect itself from ‡ooding. The success in holding back the storm waters will depend on the minimum height or strength of the di¤erent sections of the dyke. As such, it is the weakest dyke that will succumb to the storm …rst, resulting in the entire island being ‡ooded. Similarly, a weakest-link problem may be observed when trying to prevent the spread of infectious diseases such as Ebola, combating the entry of illegal drugs into a country, or controlling illegal immigrants. The weakest link model is also relevant for pollution controls.
Consider a global pollutant such as CO 2 where the damage to the environment depends on total emissions. If industries can relocate easily, then the level of pollution would depend on the country with the weakest environmental regulation.
Our analysis shows that although both models have multiple Nash equilibria, when ambiguity is su¢ ciently high, Equilibrium Under Ambiguity (henceforth EUA) is unique. Ambiguityaversion will cause people to choose the highest e¤ort level in the Best-Shot model and the lowest in the weakest link model. We proceed to test our results in the laboratory. Our experimental hypothesis is that ambiguity will decrease individuals'contributions in the weakest-link model, whereas it will increase them in the best-shot case. Kilka and Weber (2001) conducted an experiment where they asked German subjects to rate their competence when judging stock price changes of Deutsche Bank (Germany's largest banking group) and Dai-Ichi Kangyo Bank (one of Japan's largest banks). A majority of their subjects (51 of the 55 studied) reported that they felt less competent when judging stock price changes of the foreign security as opposed to the domestic security.
We test whether players in games feel a similar lack of competence when dealing with foreign opponents. If the analogy were to hold, a subject would feel more anxious when faced by a foreign opponent, than when he is faced by a local one. The rationale behind this hypothesis is that he believes the local opponent has been raised in a similar socio-cultural background as himself. Thus the behaviour of a foreign opponent, about whom there is limited knowledge, is less predictable.
We …nd that behaviour of the subjects is consistent with our hypothesis and that ambiguity does indeed lead subjects to decrease (resp. increase) contributions in the weakest-link (resp. best-shot) game. However, though subjects display ambiguity aversion on the whole, the level of ambiguity does not become more pronounced when they are matched against a foreign opponent.
Organisation of the Paper In Section 2 we describe our framework and de…nitions. The public goods models are analysed theoretically in Section 3. In Section 4 we describe our experimental design and the next section discusses the data we …nd. Section 6 compares our …ndings with related literature and Section 7 provides a summary of our results together with future avenues of research.

Framework and De…nitions
In this section we explain how we represent ambiguity in public good models. If a public good is provided by voluntary contributions the individuals concerned are playing a non-cooperative game. The pay-o¤ of any given individual will depend on the contributions of all other individuals as well as his/her own contribution. Thus to understand the impact of ambiguity we need a theory of ambiguity in games.
In a Nash equilibrium, players behave in a manner that is consistent with the actual behaviour of their opponents. Players can perfectly anticipate the actions of their opponent and can thus choose a best response to it. If beliefs are non-additive, we need to modify the idea of having consistent beliefs and the ability to play a best response. We assume that players choose pure strategies. In equilibrium, the support of a player's beliefs must be best responses for the opponent, given his/her beliefs.
Our notation for games is as follows. A 2-player game = hf1; 2g ; X 1 ; X 2 ; u 1 ; u 2 i consists of players, i = 1; 2, …nite pure strategy sets X i and payo¤ functions u i (x i ; x i ) for each player.
Both players have the same strategy set, X 1 = X 2 N; which consists of all integers between a lower bound x and an upper bound x: 1 The notation, x i ; denotes the strategy chosen by i's opponent and the set of all strategies for i's opponent is X i : The space of all strategy pro…les is denoted by X: We shall adopt the convention that female pronouns (she, her etc.) denote player 1 and male pronouns denote player 2. 2 A player has a possibly ambiguous belief about what his/her opponent will do. These beliefs are represented by capacities, which are similar to subjective probabilities except that they are not required to be additive over disjoint events. Formally capacities are de…ned as follows.
De…nition 2.1 A capacity on X i is a real-valued function on the subsets of X i such that A B ) (A) 6 (B) and (?) = 0; (X i ) = 1: The expected payo¤ obtained from a given act, with respect to a non-additive belief, i ; can be found using the Choquet integral, de…ned below.
De…nition 2.2 The Choquet integral of u i (x i ; x i ) with respect to capacity i on X i is: where the strategy pro…les in X i are numbered so that u i 1 The restriction to integer values enables us to apply the equilibrium concept from Eichberger and Kelsey (2014). It is not essential to require e¤ort levels to be integers. It could be any other …nite set of real numbers: 2 Of course this convention is for convenience only and bears no relation to the actual gender of subjects in our experiments. Schmeidler (1989) axiomatised preferences which may be represented by maximising a Choquet integral with respect to a capacity. Chateauneuf, Eichberger, and Grant (2007) introduced the neo-additive capacity model, which is a special case of Schmeidler's theory. In this model, the decision-maker has beliefs based on an additive probability distribution i : However, these beliefs are ambiguous. The con…dence in the belief is modelled by (1 i ); with i = 1 corresponding to complete ignorance and i = 0 denoting no ambiguity. The decision-maker's attitude to ambiguity is measured by i : The higher the i ; the more ambiguity-averse the decision-maker will be. These preferences are de…ned as follows.
Chateauneuf , Eichberger, and Grant (2007) show that the Choquet expected value of a pay-o¤ function u i (x i ; ) with respect to the neo-additive-capacity i is given by: where E i denotes conventional expectation with respect to the probability distribution i : This expression is a weighted average of the highest payo¤, the lowest payo¤ and an average payo¤. The response to ambiguity is partly optimistic represented by the weight given to the best outcome and partly pessimistic. We de…ne the support of a neo-additive capacity to be the support of the additive probability on which it is based. 4 De…nition 2.4 The support of a neo-additive capacity i (Aj i ; is given by supp i = supp i : 3 Where convenient we shall suppress the arguments ; and and simply write (A) : Chateauneuf, Eichberger, and Grant (2007) write the neo-additive capacity in the form (A) = +(1 ) (A) : In the main text we have modi…ed the de…nition of a neo-additive capacity to be consistent with the majority of the literature where is the weight placed on the minimum expected utility. 4 For a justi…cation of this de…nition and its relation to other support notions see Eichberger and Kelsey (2014).
In games, i is determined endogenously as the prediction of the players from the knowledge of the game structure and the preferences of others. In contrast, we treat the degrees of optimism, i and ambiguity, i ; as exogenous. De…ne the best-response correspondence of player i given that his/her beliefs are represented by a neo-additive capacity i by R De…nition 2.5 (Equilibrium under Ambiguity) A pair of neo-additive capacities In equilibrium, strategies in the support of a player's belief must be best responses for his/her opponent, given the opponent's belief. However, each player lacks con…dence in his/her likelihood assessment and responds partly in an optimistic way by over-weighting the best outcome and partly in a pessimistic way by over-weighting the worst outcome. 5 Equilibrium strategies are given by the supports of the capacities, which are required to be best-responses. If these are singleton sets, we have a pure equilibrium. Otherwise we shall say that an equilibrium is mixed. In the present paper we only consider pure equilibria. Even without ambiguity, mixed strategy equilibria are di¢ cult to interpret. Since the games studied in the present paper always have pure equilibria, we do not need to consider mixed equilibria.

Best Shot
In this section we study the e¤ect of ambiguity in the Best-Shot public goods model. If players are ambiguity-averse, we show that when ambiguity is su¢ ciently high, both of them will choose the highest possible strategy. To understand this, recall that in Nash equilibrium, one player will provide the highest possible e¤ort level and the other will free-ride by supplying the lowest e¤ort. However with high ambiguity each player becomes concerned about the worst scenario, which is his/her opponent supplying low e¤ort. This can cause both players to choose the highest possible e¤ort level. In this case ambiguity encourages the individuals to supply more e¤ort. These e¤ort levels would be ine¢ ciently high in the corresponding situation without ambiguity. It is less clear that they are ine¢ cient when ambiguity is present, since they do protect the agents against ex-ante utility losses due to ambiguity-aversion.
If both players are ambiguity-loving, then one may get an equilibrium where each provides the lowest possible e¤ort level. Ambiguity-loving causes a player to over-weight the best outcome, which occurs when the opponent chooses the highest e¤ort. This over-weighting reduces the given player's perceived marginal bene…t, which results in the choice of the lowest e¤ort level. If one player is ambiguity-loving and the other is ambiguity-averse then we can get an equilibrium where the ambiguity averse player uses the highest strategy and the other player uses the lowest strategy. As above, ambiguity-aversion increases the incentive to choose a high strategy and ambiguity-loving increases the incentive to play a low strategy.
Finally if ambiguity is low then there are multiple EUAs. In one Player 1 plays the highest strategy and Player 2 plays the lowest strategy and in the other the roles are reversed. This is similar to the standard Nash equilibria. 4. if (1 1 ) + 1 1 > c > 1 1 and (1 2 ) + 2 2 > c > 2 2 ; there are two possible pairs of equilibrium strategies h x; xi and hx; xi : 5. if 1 1 < c and 2 2 > c; the equilibrium strategies are unique and are equal to hx; xi : 6. if 1 1 > c and 2 2 < c; the equilibrium strategies are unique and are equal to h x; xi : 7. if 1 1 > c and 2 2 > c; the equilibrium strategies are unique and are given by, Proof. Consider Player 1: Suppose that she believes that Player 2's e¤ort will bex 2 ; where x 6x 2 6 x: Then 1's (Choquet) expected pay-o¤ conditional on this belief will be: By similar reasoning the pay-o¤ of Player 2 is given by: Part 1 Clearly this implies 1 1 c < 0: Hence, by equation (1), V 1 (x 1 jx 2 ) is a strictly decreasing function of x 1 : The only possible best response is x 1 = x: Similarly the best response for Player 2 is x and thus the equilibrium strategies are hx; xi : Part 2 As in part 1, V 1 (x 1 jx 2 ) is a strictly decreasing function of x 1 and the only possible best response is x 1 = x: From equation (2) we see that V 2 (x 2 jx) is increasing in x 1 : Hence Player 2's best response is x and thus the equilibrium strategies are hx; xi :

Part 3
By similar reasoning to that used in part 2, we may show that h x; xi is the unique equilibrium strategy pro…le.

Part 4
By equation (1), V 1 (x 1 jx 2 ) is decreasing for x 1 <x 2 and increasing for x 1 >x 2 : Consequently the only possible best responses are x 1 = x or x 1 = x: Since Player 2 is in a similar position ifx 2 is a best response we must havex 2 = x orx 2 = x: Ifx 2 = x; then is decreasing, hence Player 1's best response is x and thus the equilibrium strategies are hx; xi : Ifx 2 = x; then V 1 (x 1 jx) is increasing, Player 1's best response is x and hence the equilibrium strategies are h x; xi :

Part 5
Note that 2 2 > c implies (1 2 ) + 2 2 > c: From equation (2) we see that that V 2 (x 2 jx 1 ) is strictly increasing in x 2 for all values ofx 1 : Hence Player 2's best response is Consequently the equilibrium strategies are hx; xi and are unique.

Part 6
Follows by similar reasoning to part 5.

Part 7
Then by parts 5 and 6 the best responses of player 1 and 2 are x 1 = x and x 2 = x respectively. This holds regardless of the player's belief about the behaviour of his/her opponent. Consequently the equilibrium strategies are h x; xi and are unique.
The following result shows that, in general, when ambiguity is su¢ ciently high the equilibrium of the best-shot game will be unique.

Weakest Link
In the weakest link model, provision of the public good is equal to the minimum of the individual contributions; see Cornes and Sandler (1996)). This model is also known as the minimum e¤ort coordination game; Huyck, Battalio, and Beil (1990). In the absence of ambiguity, any situation where both players choose the same e¤ort level is a Nash equilibrium.
Below we show that if there is su¢ cient ambiguity, then equilibrium will be unique. If players are su¢ ciently ambiguity-averse, they will both use the lowest possible strategy. In contrast, they will choose the highest possible strategy if they are su¢ ciently ambiguity-loving. For lower degrees of ambiguity the EUA is similar to the equilibrium without ambiguity. Coordinating on any of the possible e¤ort levels constitutes an equilibrium. The intuition is that ambiguityaversion can cause a given individual to be concerned that somebody else will make a low contribution. In which case his/her own e¤ort will be wasted. As a result the given individual will make a low contribution. However if all think likewise, the only equilibrium is where everybody makes the smallest possible contribution.
The following result describes the impact of ambiguity in the weakest link public good model. As one might expect increases in ambiguity-aversion make it more likely that a player will provide the lowest e¤ort. If there is a high degree of ambiguity and players are very ambiguity loving, then it is possible that both will choose the highest possible strategy. However we believe that this outcome is unlikely to be observed in practice. For low levels of ambiguity there are multiple EUA. In contrast for high degrees of ambiguity the equilibrium is unique. 2. if 1 1 1 > c > 1 (1 1 ) and 1 2 2 > c > 2 (1 2 ) ; then hx 1 ;x 2 i is a pair of EUA strategies if and only ifx 1 =x 2 ; 3. if 1 (1 1 ) > c and 1 2 2 > c; the equilibrium strategies are unique and are equal to h x; xi; 4. if 1 1 1 > c and 2 (1 2 ) > c; the equilibrium strategies are unique and are equal to h x; xi.
Proof. Consider Player 1: Suppose that she believes that Player 2's e¤ort will bex 2 ; where x 6x 2 6 x: Then 1's (Choquet) expected pay-o¤ conditional on this belief will be: Similarly if Player 2 believes that Player 1's e¤ort will bex; then 2's (Choquet) expected pay-o¤ will be: Part 1 Clearly 1 1 1 > 1 1 1 : If c > 1 1 1 then V 1 is strictly decreasing in x 1 : Thus Player 1 has a unique best response, x: Given this, Player 2's (Choquet) expected pay-o¤ is decreasing in x 2 ; hence his best response is x: The proof for the case where c > 1 2 2 is similar.

Part 2
Let hx 1 ;x 2 i be a pair of EUA strategies. Suppose, if possible, thatx 1 >x 2 : Then by equation (3), V 1 is strictly decreasing in x 1 for x 1 >x 2 : This establishes thatx 1 is not a best response, contrary to the original hypothesis. Thus we must havex 1 6x 2 : A similar argument establishes thatx 2 >x 1 ; which implies thatx 2 =x 1 : To prove the converse assume that x 0 2 X: We need to show that x 0 1 = x 0 2 = x 0 are EUA strategies. Substitutingx 2 = x 0 2 into equation (3) we …nd that V 1 is increasing in x 1 for x 6 x 1 6 x 0 and decreasing in x 1 for x 0 6 x 1 6 x: This implies that x 1 = x 0 1 is Player 1's best response. By similar reasoning x 2 = x 0 2 is a best response for Player 2: It follows that hx 0 1 ; x 0 2 i is a pair of equilibrium strategies.

Part 3
Note that 1 (1 1 ) > c implies 1 1 > c; hence V 1 is strictly increasing in x 1 for any beliefs she has about Player 2: Thus x 1 = x is her best response. Given thatx 1 = x; V 2 is strictly increasing in x 2 ; which implies that Player 2's best response is x 2 = x: Hence there is a unique pair of equilibrium strategies, h x; xi :

Part 4
Follows by similar reasoning to part 3.
The intuition for this result is as follows. In part 1 a given player i, perceives a high degree of ambiguity and is ambiguity-averse thus i i is close to 1. This causes him to overweight the possibility that his opponent will supply minimal e¤ort. He decides to supply minimal e¤ort since anything more is wasted. His opponent responds by also supplying low e¤ort. Parts 3 and 4 describe a situation where one individual perceives a high degree of ambiguity and is also extremely ambiguity-loving. This causes him/her to choose the highest possible strategy.
His/her opponent chooses the highest possible strategy in response.
Part 2 is similar to the situation without ambiguity, since any situation in which both players choose the same e¤ort level can be an equilibrium. However as ambiguity increases, the range of parameter values in this case shrinks and eventually it disappears altogether. One may show in general the equilibrium is unique provided ambiguity is su¢ ciently high.
In this section we have shown that ambiguity-aversion tends to reduce contributions in the weakest link model but increases them in the best shot case.

Experimental Model
In this section we describe the games used in our experiments and the predicted behaviour. We aim to test the hypothesis that ambiguity has opposite e¤ects in games of strategic complements and substitutes; see Eichberger and Kelsey (2002).
The task given to the subjects was to choose an e¤ort level from the set X = f100; :::; 150g. By Proposition 3.2, the Nash equilibrium of the weakest-link game is for both players to coordinate on any one of the six e¤ort levels available, thus f(x 1 ; x 2 ) 2 X j x 1 = x 2 g. As a result, there are multiple Nash equilibria. Given this, it is understandable that there would be ambiguity among the subjects about which e¤ort level they should attempt to coordinate on.
The equilibrium action with a high level of ambiguity would be for a subject to choose an e¤ort level of 100, which gives him an ambiguity-safe payo¤ of 50ECU (See Figure 2); irrespective of his opponent's choice: Selecting an e¤ort level of 100 thus frees the subject from having to depend on his opponent's choice and/or having to achieve perfect coordination in their chosen e¤ort levels.
By Proposition 3.1, the best-shot game has two pure Nash equilibria: 6 hx 1 ; x 2 i = h100; 150i and hx 1 ; x 2 i = h150; 100i : Nash equilibrium predicts that one of the players will exert the highest e¤ort level (in our case 150) ; while the other will free-ride and choose the lowest e¤ort available (in our case 100). Here again, we have multiple Nash equilibria and it is expected that subjects would perceive ambiguity about which one to choose. If the level of ambiguity about the opponent's choice is high, the equilibrium action under ambiguity is to choose the highest e¤ort level, i.e., 150; since this provides the player with a constant payo¤ irrespective of the opponent's decision. Treatments I and III consisted of 60 subjects each and Treatment II had 61 subjects. In total there were 181 subjects who took part in the experiment, 81 of whom were males and the remaining 100 were females. We were interested in whether participants had a quantitative background -59 of the subjects had studied a quantitative subject such as Biochemistry, Electronic Engineering or Astrophysics, while 122 of the subjects had studied a non-quantitative subject such as History, Philosophy, or International Relations. Each experimental session lasted a maximum of 45 minutes including payment.

Experimental Design
Subjects …rst read through a short, comprehensive set of instructions at their own pace, following which the instructions were also read out to all the participants in general. 8 The subjects were asked to …ll out practice questions to check that they understood the games correctly. Each subject was then asked to choose one e¤ort level for the weakest-link game, followed by an e¤ort level for the best-shot game. Subjects played each game exactly once.
We believe that this made the games one-shot, as subjects received no feedback between the rounds and could not incorporate any learning.
Once subjects had made both decisions, a throw of dice determined one game round for which they would be paid. We picked one round at random for payment in order to prevent individuals from self-insuring against payo¤ risks across rounds; see Charness and Genicot (2009). If all rounds count equally towards the …nal payo¤, subjects are likely to try and accumulate a high payo¤ in the …rst few rounds and then care less about how they decide in the following rounds. In contrast, if subjects know that they will be paid for a random round, they treat each decision with care.
Players'decisions were matched according to a predetermined random matching, and pay-o¤s were announced.

Data Analysis and Results
Treatment I In this treatment, subjects were matched against other locally recruited subjects only. 9 In the weakest link experiment, we …nd that 22% (13) of the subjects chose an e¤ort level of 100; (See Figure 3): This is the e¤ort level at which the subject has a constant payo¤, which is independent of the opponent's action. Moreover, 65% (39) of subjects chose an e¤ort level between 100 and 120, i.e., the lower end of the spectrum of e¤ort choices. This con…rms the theoretical prediction that ambiguity would lead to subjects reducing their e¤ort levels. Some subjects (9); chose the maximum e¤ort level, 150; however, they were in a very small minority.
Previous experiments conducted by us, Kelsey and le Roux (2015), have found that a minority of subjects display ambiguity-seeking behaviour in Ellsberg-urn type decision problems. Thus, the present experiment con…rms that similar ambiguity-seeking behaviour is also observed in games. In the best-shot round, we …nd that 47% (28)   While analysing the manner in which people switch e¤ort levels between the two scenarios, we …nd that 55% (33) of our subjects switched from a low e¤ort level in the weakest-link round, to a higher e¤ort level in the best-shot game (See Table 1). These subjects display ambiguityaverse behaviour, which is in line with Eichberger and Kelsey (2002). Interestingly, we …nd that 25% (15) of subjects display a preference for ambiguity, choosing a high e¤ort level in the weakest-link game and then switching to a low e¤ort level in the best-shot round. We also note that 20% (12) of subjects did not change their chosen e¤ort levels between the two rounds -these subjects could be displaying ambiguity neutral behaviour. 10 Treatment II In this treatment, Exeter subjects were matched with the foreign opponent only. In the weakest-link round, only 8% (5) of the subjects chose e¤ort level 100 (See Figure   4). Even though the constant-payo¤ e¤ort level has been chosen by a small minority, a sizeable 59% (36) of subjects have chosen low contribution levels in the range 100 120. In the bestshot game, 43% (26) of the subjects chose an e¤ort of 150, which is the equilibrium action under ambiguity, while 59% (36) of the subjects chose high contribution levels in the range of 130 150. We see this as an indication that ambiguity is resulting in e¤orts being concentrated at the lower end of the available set of e¤orts in the case of the weakest-link game; and at the higher end, in case of the best-shot game (See Figure 4): Table 1, summarises the manner in which people switch e¤ort levels between the two scenarios. We …nd that 46% (28) of the subjects who choose a low e¤ort level in the weakest-link experiment switch to a higher e¤ort level in the best-shot game. This is compatible with ambiguity-averse behaviour. Moreover, 33% (20) of subjects display ambiguity-seeking behaviour, choosing a high e¤ort level in the weakest-link round followed by a lower e¤ort level in the best-shot game, while 21% (13) of subjects do not change their chosen e¤ort levels between the two rounds. It is interesting to note from Table 1, that more subjects displayed ambiguity-seeking behaviour when faced by the foreign subject (compared to Treatment I). We had expected subjects to be more ambiguity-averse/feel more anxious when matched against the foreign subject, but our data pointed in the opposite direction. On the whole, 58% (35) of the subjects chose a low e¤ort level between 100 120, against the local opponent, while 53% (32) chose an e¤ort in that range against the foreign opponent.
We …nd (as in Treatment II) fewer people behaving in an ambiguity-averse manner against the foreign subject.
In the best-shot game, we …nd that 40% (24) of the subjects chose the maximum e¤ort 150 against the local opponent, while 43% (26) chose it against the foreign opponent. Moreover, 55% (33) and 57% (34) of subjects chose in the high e¤ort range of 130 150, against the local and foreign opponents respectively. We …nd that 28% (17) and 35% (21) of subjects chose e¤ort level 100 against the local and foreign opponents respectively, in the hope of freeriding. Intuitively, these subjects are over-weighting the probability of a good outcome (that their opponent has chosen e¤ort level = 150), and are thus displaying an optimistic attitude to ambiguity. Again, it can be noted that more subjects are displaying this optimistic attitude toward uncertainty against the foreign opponent! Even though we do not see a huge disparity in the e¤ort choices versus the local and foreign opponent, Figure 5 shows that ambiguity does explain (most of) the deviations from Nash equilibrium. In the case of the weakest link game, most responses are concentrated towards the lower end of the spectrum between 100 120; while in the best shot case, responses are concentrated towards the high end, i.e., at 150.

Figure 5: Treatment III -E¤ort Levels vs. Both Local & Foreign Subjects
We ran McNemar's test to check whether subjects responded to ambiguity in a pessimistic way. The null hypothesis was that the probability of choosing a low e¤ort in the weakestlink round, followed by a high e¤ort in the best-shot round (henceforth, labelled ambiguityaverse behaviour/decision), equalled the probability of choosing a high e¤ort in the weakest-link round, followed by a low e¤ort in the best-shot round (henceforth, labelled ambiguity-seeking behaviour/decision). The alternative was that the number of ambiguity-averse decisions was not equal to the number of ambiguity-seeking decisions. The statistical data follows a chisquared distribution with one degree of freedom. When looking at the data as a whole (all three treatments/nine sessions), we reject the null at a 1% level of signi…cance. The number of ambiguity-averse decisions made, on the whole, was twice the number of ambiguity-seeking decisions (See Table 2). When looking at the di¤erent treatments individually, we reject the null at a 1% level of signi…cance for Treatment I, and at a 5% level of signi…cance for Treatment II. We fail to reject the null for Treatment III, for decisions against both the local subject and the foreign subject. The McNemar Test thus demonstrates that on the whole, there was a di¤erence between the number of ambiguity-averse and ambiguity-seeking choices made. In order to investigate further, whether the number of ambiguity-averse decisions exceeded the number of ambiguity-seeking decisions, we ran a Wilcoxon signed-ranks test. The null hypothesis was that the number of ambiguity-averse decisions made equalled the number of ambiguity-seeking decisions; while the alternative was that ambiguity-averse behaviour exceeded ambiguity-seeking behaviour. We ranked behaviour in all the di¤erent sessions, and found that we could reject the null at a 1% level of signi…cance. Furthermore, if we consider a more simple sign test, with the same null and alternative as before, we …nd that we can reject the null at a 5% signi…cance level: Thus, we …nd that subjects showed ambiguity-averse behaviour signi…cantly more often in our experiments, than ambiguity-seeking behaviour.

Related Literature
Experiments on the weakest-link game were previously studied by Huyck, Battalio, and Beil (1990). They study tacit coordination in this game, and conclude that it is unlikely that a payo¤-dominant equilibrium would be chosen in a one-shot game or in repeated play. Moreover, they …nd that when there are a large number of players attempting to coordinate, the equilibrium is secure but ine¢ cient. Our results in the weakest-link round are consistent with their conclusions. We …nd that 59% (142) of subjects chose an e¤ort level in the range 100 120, which would result in a payo¤-dominated equilibrium. Furthermore, even though our game consisted of only two subjects coordinating (and not a large number of players), we found that 21% (51) of subjects chose an e¤ort level of 100, which would have resulted in a secure but ine¢ cient equilibrium. Harrison and Hirshleifer (1989), compare contributions to a public good in a sealed bid as well as a sequential game. They use a repeated game scenario which implements all three possible versions of the game -standard summation, weakest-link and best-shot, in order to ascertain which of the three formats results in the greatest free-riding. They …nd that both sealed bid as well as the sequential game treatments, con…rmed their hypothesis that the under-provision of the public good expected under the standard format, is mitigated under the weakest-link format, but aggravated under the best-shot version. In contrast we found that 55% (33) and 46% (28) of subjects in Treatment I and II switched from a low e¤ort level in the weakest-link round to a higher e¤ort level in the best-shot game (See Table 1). This may be because the one-shot nature of our games increases ambiguity and prompts players to increase contributions. Goeree and Holt (2001) (henceforth GH) study, a set of games which initially conform to Nash predictions when tested experimentally. However, they show that in each case a change in a parameter, which is not relevant according to Nash equilibrium leads to a large change in observed behaviour and failure of Nash predictions. In particular, they study the weakest link model where subjects could choose an e¤ort from the set E = f110; :::; 170g at a marginal cost of either c = 0:1 or c = 0:9: Recall that in Nash equilibrium subjects coordinate on the same e¤ort level f(e 1 ; e 2 ) 2 E 2 j e 1 = e 2 g : GH …nd that for low marginal costs (c = 0:1), subjects choose high e¤ort levels and for high marginal costs (c = 0:9) ; a majority of subjects choose low e¤ort levels. They conclude that this concentration of choices at the lower (resp. higher) end of the e¤ort spectrum is caused by the high (resp. low) marginal cost of e¤ort. Eichberger and Kelsey (2011) argue that GH's results can be explained by ambiguity.
In our experiment we chose a value of marginal cost, (c = 0:5), between the two values used by GH. We …nd that an intermediate proportion of subjects (22%) choose the lowest e¤ort level. We …nd that subjects'e¤ort choices depend on the e¤ect of ambiguity, given the strategic nature of the game being played. This can be seen in Table 1, where even though the marginal cost of the e¤ort is constant at c = 0:5; subjects switch their e¤ort levels depending on whether it is the weakest-link or the best-shot game being played.

Conclusions
In this paper we have developed a theoretical model of the e¤ect of ambiguity in the best shot and weakest-link models of public good provision. We then proceeded to test the theory experimentally. Subject behaviour was found to be consistent with our hypothesis. We …nd that in the presence of ambiguity, subjects choose low e¤ort levels in the weakest-link game and high e¤ort levels in the best-shot game. Moreover, we …nd that on average, 51% (61) of the subjects who took part in Treatments I and II, display ambiguity-averse behaviour; 29% (35) of subjects display ambiguity-seeking behaviour; and 20% (25) of subjects do not change their chosen e¤ort levels between the two rounds. 12 We expected the subjects to display a greater level of ambiguity-averse behaviour when faced by a foreign opponent. However, although we observe ambiguity-averse behaviour on the whole, we …nd a signi…cant minority of subjects who display an optimistic attitude towards ambiguity. This is quite a curious …nding, as one would expect that the ambiguity-safe option would be chosen more often against the foreign subject. Our …ndings contrast with those of Kilka and Weber (2001), who found that subjects are more ambiguity-averse when the returns of an investment are dependent on the performance of foreign securities than when they are linked to domestic securities.
There may be several other reasons that might explain why the level of ambiguity when facing a foreign subject may remain unchanged. One may be that the subjects wanted to be consistent in their choices. If this was the case, they would put extra e¤ort into choosing the same action against both opponents. In addition, the returns of a bank depend in a complex way on …nancial markets and hence present a more di¢ cult decision problem than …nding the right strategy in a game with a relatively small strategy space. This may explain part of the heightened "ambiguity" captured by Kilka and Weber (2001), where the subjects were presented with the option of an investment dependent on foreign securities. It is easier for subjects to conceptualise another person whom they may be faced with, than investments in known/unknown …nancial markets. It may be interesting to run a follow-up experiment, where subjects are given the choice of either facing a foreign opponent or investing in a foreign security.
Moreover, it may have been the case that subjects viewed the foreign student as akin to any other local student. This is not that di¢ cult to understand. Globalisation and increased media awareness, together with the spreading tentacles of social networking and escalating international student numbers, have ensured that a foreign subject (in this case from India) is not an unknown quantity any more. There are not many parts of the world, that hold the kind of ambiguity for us today, as there were in the past.