OWNER-INTRUDER CONTESTS WITH INFORMATION ASYMMETRY

. We consider kleptoparasitic interactions between two individuals - the Owner and the Intruder - and model the situation as a sequential game in an extensive form. The Owner is in possession of a resource when another individual, the Intruder, comes along and may try to steal it. If the Intruder makes such a stealing attempt, the Owner has to decide whether to defend the resource; if the Owner defends, the Intruder can withdraw or continue with the stealing attempt. The individuals may value the resource diﬀerently and we distinguish three information cases: (a) both individuals know resource values to both of them, (b) individuals know only their own valuation, (c) individuals do not know the value at all. We solve the game in all three cases. We identify scenarios when it is beneﬁcial for the individuals to know as much information as possible. We also identify several scenarios where knowing less seems better as well as show that an individual may not beneﬁt from their opponent knowing less. Finally, we consider the same kind of interactions but without the option for the Intruder to withdraw. We ﬁnd that, surprisingly, the Intruder typically fares better in that case. [1, 3, 5, 6, 20]. Many recent models contain a high degree of detail and realism, see for example [21, 22, 23, 48, 49]. 7 An Owner-Intruder game is a common way to model kleptoaparasitic interactions; it can account for many 8 diﬀerent situations and assumptions while yielding clear and testable predictions [12, 14, 16, 17, 19, 36], see 9 also [41] or [26] for recent reviews.


Introduction 1
In nature, kleptoparasitism, the act of stealing and fighting over resources, is common among birds [43, 44, 46]; 2 but it occurs across many types of species such as insects [29], fish [24] and mammals [31]. There are many 3 different behaviors involved in kleptoparasitic interactions. Sometimes individuals will defend and fight for the 4 resource and sometimes the resource is simply forfeited without any conflict [27]. 5 There are many models of kleptoparasitic interactions with varying degree of complexity, see for example 6 [1, 3, 5, 6, 20]. Many recent models contain a high degree of detail and realism, see for example [21,22,23,48,49].
fighting over territory [39]. Second, individuals may know how much they value the resource themselves, but 23 do not know the value to the opponent. This is the case when RHP cannot be easily assessed, for example 24 in the contests between Mozambique mouthbrooder, Oreochromis mossambicus, [47], or damselflies, Calopteryx 25 maculata, [34]; see [35] for more details. Finally, individuals may not know the resource value for themselves 26 or for their opponent. Such scenarios are likely not common, but may theoretically still happen among animals 27 like the great tit, Parus major, [30] when the contest happens between the intruder and a new territory owner 28 that does not yet know the true value of the territory. 29 In this paper, we consider a scenario where one individual, the Owner, has a valuable resource that another 30 individual, the Intruder, may want to steal. In Section 2 we set up the Owner-Intruder game as an extension 31 of the game considered in [9]. In Section 3 we solve the game for all three information cases. In Section 4 we 32 compare the outcomes between the cases to see the effect of extra information. In Section 5 we investigate the 33 differences in outcomes between [9] and our extended model. We conclude our paper by a discussion in Section 34 6. 35

36
We model the conflict between the Owner and the Intruder as a sequential game in extensive form. Our 37 model extends the game studied in [9] by one round (in which the Intruder can withdraw from the conflict or 38 continue with the stealing attempt by attacking the Owner), see Figure 1. The notation is summarized in Table   39 1.

40
The Owner is in possession of a resource when another individual, the Intruder comes along and may try to 41 steal it. If the Intruder makes such a stealing attempt, the Owner has two options: O1) it can either flee the 1 − a. The winner will gain the resource, and the loser will gain nothing. Both individuals will have to pay a 48 cost c for the fight.  Throughout this paper, we will assume that V O and V I are independent identically distributed random    Consequently, the results from [9] can be readily adapted to the current game. The behavioral outcomes and corresponding payoffs are summarized in Table 2, see also Figure 2. It follows from Table 2 The Owner flees The Owner defends The Intruder withdraws The Owner defends The Intruder attacks The Owner flees The Owner defends The Intruder withdraws The Owner defends The Intruder attacks The Owner flees The Owner defends The Intruder withdraws The Owner defends The Intruder attacks (c) No information case

Behavior and Payoffs Full information Partial information No information The Owner The Intruder
Defends Withdraws Table 2. Summary of behavioral outcomes and payoffs depending on the information case and conditions on V I and V O . In all cases, the Intruder tries to steal first. We note that this table could be reconstructed directly from Table 13.1 in [9] and vice versa by the following substitutions: "The Owner" ↔ "Scrounger", "The Intruder" ↔ "Producer", "V O " ↔ "ν s ", "V I " ↔ "ν p ", "a" ↔ "1 − a", "π I " ↔ "π".

Comparison between different information cases 69
In this section we provide the comparison between the full, partial, and no information cases for the Owners Owner (resp. the Intruder) in the full, partial, and no information case will be denoted P F O , P P O , P N O (resp. 72 P F I , P P I , P N I ). 73 We note that V O and V I are random variables and the payoffs depend on V O and V I . While we can (and 74 often will) compare the payoffs for specific values of V O and V I , we will also compare the mean values of the 75 payoffs (as in Figure 3). We will also explore the distribution of the payoff values in Figures 4 and 5. For all three information cases, the mean payoffs to the Owner and the Intruder are shown in Figure 3. The

101
distribution of the payoffs are demonstrated in Figure 4 for the Owner and in Figure 5 for the Intruder. We 102 note that all payoffs are non-negative in the full information case. The Owner may have a negative payoff in 103 the partial information case and the no information case (although the mean payoffs are both non-negative).

104
The Intruder has non-negative payoffs in the partial information case and can have negative payoffs (but with 105 non-negative mean) in the no information case. In this section, we compare our results with the results of [9] where the Intruder did not have the option 108 to withdraw before the conflict escalates. Intuitively, one would think that adding this option will only be 109 beneficial to the Intruder, but we will see that it is not always the case. The results of [9], in our current 110 terminology and notation, are summarized in Figure 6.    The Intruder tries to steal The Intruder does not steal The Intruder tries to steal The Owner defends The Owner flees The Intruder tries to steal The Intruder does not steal The Intruder tries to steal The Owner defends (c) No information case

158
Not surprisingly, under most circumstances, it is beneficial for the individual to know more rather than to 159 know less. In particular, Owner's payoff in the full information case is larger than in the partial information case. Metellina mengei fight over females [2] and the Owner may know the value of the reward to itself, but the 173 Intruder does not; such a situation was not captured by our current model. 174 We saw that increasing the opponent's knowledge may be helpful in some instances and detrimental in others.

175
Specifically, contestants prefer opponents to know that they are willing to fight. They also prefer to hide that 176 they are not going to fight when challenged. This may be the case of bald eagles contesting over a prey item, 177 who often assess the size and hunger level of their opponents and attack those most likely to retreat [25]. In 178 general, the fact that an individual may benefit from an opponent's knowledge may be a factor behind the 179 evolution of signalling, see for example [38]. 180 We also studied the effect of the extra round during which the Intruder can withdraw before the contest 181 escalates. We saw that, surprisingly, this round had an adverse effect on the Intruder's payoff in the full and the fight is costly also for the Owner who thus flees and no fight will take a place. Yet, when the Intruder can 185 withdraw, the Owner can call the bluff and display its willingness to defend, effectively forcing the Intruder to 186 give up and withdraw. When the fight cost is not so high, the extra round has no effect on the interaction. The