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. Mathematics Subject Classification. 91A05, 91A40, 91A22. Received February 3, 2020. Accepted January 16, 2021.

Cost of the fight a Probability of the Owner winning the fight π I Probability that the Intruder will attack; π I = P rob c 1−a < V I π O Probability that the Owner will defend; π O = P rob c a < V O P Info case O Payoff to the Owner in the given information case P Info case I Payoff to the Intruder in the given information case Expected value of the resource for the Owner or Intruder of the resource to the Owner and V I the value of the same resource to the Intruder. We will 56 distinguish three cases: 1) the full information case when the Owner and the Intruder both know

Analysis
In the full information case, both the Owner and the Intruder know V O and V I . The Owner 69 has two options -defend the resource or flee the area and give up the resource. The payoff to the 70 Owner depends on the Intruder's action. So, we first need to find Intruder's optimal behavior.

71
If the Owner flees, the Intruder does not need to decide anything, it simply takes the resource.

72
In this case, the Owner gets 0 and the Intruder gets V I .
If the Intruder does not attack, its payoff will be 0. If the Intruder attacks, the individuals will

99
The behavioral outcomes and corresponding payoffs are summarized in Table 2, see also Figure   100 2. It follows from the Table 2 that the payoffs to the individuals depend on the relationship between

Behavior and Payoffs Full information Partial information No information Owner Intruder
Defends Flees

Flees
Takes over Now, we will analyze the decision of the Owner. Let π I be the probability that the Intruder is 109 going to attack. By, (3), we get The Owner has two options, to defend or to flee. If it flees, it will get 0. If the Owner defends, ( Consequently, the Owner should defend only if The behavioral outcomes and corresponding payoffs are summarized in Table 2, see also Figure   118 2. In this section we assume that neither the Owner nor the Intruder knows the value of the 121 resource (neither for themselves, nor for their opponent). They do know, however, that the expected 122 value of the resource for either one of them is E[V ].

123
The analysis follows the steps in Section 3.1. The only difference is that here we need to use we will keep the distinction between the two to make it clearer which 126 condition relates to which decision.

127
As in Section 3.1, the Intruder should attack only if 0 and the Owner should defend only if aE The behavioral outcomes and corresponding payoffs are summarized in Table 2, see also Figure   130 2.

Comparison between different information cases 132
In this section we provide the comparison between the full, partial, and no information cases for 133 Owners and Intruders. The summary is provided here, the details are then provided in Section 4.1 134 and Section 4.2. The payoffs to the Owner (resp. Intruder) in the full, partial, and no information 135 case will be denoted P F O , P P O , P N O (resp. P F I , P P I , P N I   First, assume that c 1−a > V I . In this case, Similarly, P F I = 0 and at the same time, P P I is either Second, assume that c 1−a < V I and In the latter case, To summarize, by considering that the above three cases exhaust all the options, we get that In particular, it is always beneficial for the Owner to know the value of V I . At the same time, Fourth, assume that c < aE[V O ]. We will distinguish three cases to understand the relationship   it is better for the Intruder to go first and it is better for the Owner to go second. for the Owner to defend their resource and for the Intruder to fight for it. 255 We observed that the actual contests occur only when the cost of the fight is relatively low 256 compared to the resource value. This is in an agreement with previous experiments. For example, 257 fights among group living pholcid spiders, Holocnemus pluche, are more common over larger (more 258 valuable) prey, without any observable increase in the fight cost (Jakob, 1994).

259
Not surprisingly, under most circumstances, it is beneficial for the individual to know more 260 rather than to know less. In particular, Owner's payoff in full information case is larger than in 261 the partial information case. Such a phenomenon was also observed before. the Intruder. In both cases though, the advantage seems to come from the opponent knowing less 272 rather than the focal individual knowing more. We believe that a more detailed model is needed 273 to make the proper distinction. 274 We saw that increasing the opponent's knowledge may be helpful in some instances and detri-275 mental in others. Specifically, contestants prefer opponents to know that they are willing to fight.

276
They also prefer to hide that they are not going to fight when challenged. This may be the case 277 of bald eagles contesting over a prey who often assess the size and hunger level of their opponents 278 and attack those most likely to retreat (Hansen, 1986). In general, the fact that an individual 279 may benefit from opponent's knowledge may be a factor behind the evolution of signalling, see for 280 example Payne and Pagel (1996). 281 We also studied the effect of the order in which the individuals take actions. We saw that the