1
|
Practical Basis of the Geometric Mean Fitness and its Application to Risk-Spreading Behavior. Bull Math Biol 2022; 84:25. [PMID: 34982263 DOI: 10.1007/s11538-021-00984-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2021] [Accepted: 11/30/2021] [Indexed: 11/02/2022]
Abstract
Temporal variations in population size under unpredictable environments are of primary concern in evolutionary ecology, where time scale enters as an important factor while setting up an optimization problem. Thus, short-term optimization with traditional (arithmetic) mean fitness may give a different result from long-term optimization. In the long-term optimization, the concept of geometric mean fitness has been received well by researchers and applied to various problems in ecology and evolution. However, the limit of applicability of geometric mean has not been addressed so far. Here we investigate this problem by analyzing numerically the probability distribution of a random variable obeying stochastic multiplicative growth. According to the law of large number, the expected value (i.e., arithmetic mean) manifests itself as a proper measure of optimization as the number of random processes increases to infinity. We show that the finiteness of this number plays a crucial role in arguing for the relevance of geometric mean. The geometric mean provides a satisfactory picture of the random variation in a long term above a crossover time scale that is determined by this number and the standard deviation of the randomly varying growth rates. We thus derive the applicability condition under which the geometric mean fitness is valid. We explore this condition in some examples of risk-spreading behavior.
Collapse
|
2
|
Ito H, Tanimoto J. Dynamic utility: the sixth reciprocity mechanism for the evolution of cooperation. ROYAL SOCIETY OPEN SCIENCE 2020; 7:200891. [PMID: 32968531 PMCID: PMC7481712 DOI: 10.1098/rsos.200891] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Accepted: 07/07/2020] [Indexed: 06/11/2023]
Abstract
Game theory has been extensively applied to elucidate the evolutionary mechanism of cooperative behaviour. Dilemmas in game theory are important elements that disturb the promotion of cooperation. An important question is how to escape from dilemmas. Recently, a dynamic utility function (DUF) that considers an individual's current status (wealth) and that can be applied to game theory was developed. The DUF is different from the famous five reciprocity mechanisms called Nowak's five rules. Under the DUF, cooperation is promoted by poor players in the chicken game, with no changes in the prisoner's dilemma and stag-hunt games. In this paper, by comparing the strengths of the two dilemmas, we show that the DUF is a novel reciprocity mechanism (sixth rule) that differs from Nowak's five rules. We also show the difference in dilemma relaxation between dynamic game theory and (traditional) static game theory when the DUF and one of the five rules are combined. Our results indicate that poor players unequivocally promote cooperation in any dynamic game. Unlike conventional rules that have to be brought into game settings, this sixth rule is universally (canonical form) applicable to any game because all repeated/evolutionary games are dynamic in principle.
Collapse
Affiliation(s)
- Hiromu Ito
- Department of International Health and Medical Anthropology, Institute of Tropical Medicine, Nagasaki University, Nagasaki 852-8523, Japan
| | - Jun Tanimoto
- Department of Energy and Environmental Engineering, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
- Department of Advanced Environmental Science and Engineering, Faculty of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
| |
Collapse
|
3
|
Bankruptcy is an inevitable fate of repeated investments with leverage. Sci Rep 2019; 9:13745. [PMID: 31551500 PMCID: PMC6760122 DOI: 10.1038/s41598-019-50237-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2019] [Accepted: 09/06/2019] [Indexed: 11/17/2022] Open
Abstract
Due to the globalization and computerization of financial and economic activities, numerous repetitive leveraged investments have become possible in stock markets and currency exchanges. In reality, repeated leveraged investments up to 100 times/day are possible via online access. With computer-aided programs, this repetition number may easily increase 1000 times/day. The possibility of bankruptcy in repeated leveraged investments has never been considered in actual practices because the probability of bankruptcy in a single investment trial is almost negligible. Here, we show that the extremely numerous repetitions have a considerable chance of bankruptcy overall, even if the probability of bankruptcy for a single investment is extremely close to zero. The exact relationship between the repetitions and the probability of bankruptcy is approximated well by n(0.63) = m, where 10n is the number of repetitions, 10−m is the bankruptcy probability of a single investment, and n(0.63) is the 63% chance of bankruptcy. Thus, extremely rare events can always lead to bankruptcy in continuously repeated investment, even if the possibility of such an event is almost null. We suggest that the avoidance measure of bankruptcy is necessary in numerous repeated investments even if a single trial is almost certain to win.
Collapse
|
4
|
Ito H. Risk sensitivity of a forager with limited energy reserves in stochastic environments. Ecol Res 2019. [DOI: 10.1111/1440-1703.1058] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Hiromu Ito
- Department of General Systems Studies The University of Tokyo Tokyo Japan
- Department of International Health Institute of Tropical Medicine, Nagasaki University Nagasaki Japan
- Department of Environmental Sciences, Zoology University of Basel Basel Switzerland
| |
Collapse
|
5
|
Ito H, Katsumata Y, Hasegawa E, Yoshimura J. The promotion of cooperation by the poor in dynamic chicken games. Sci Rep 2017; 7:43377. [PMID: 28233837 PMCID: PMC5324166 DOI: 10.1038/srep43377] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Accepted: 01/23/2017] [Indexed: 11/09/2022] Open
Abstract
The evolution of cooperative behavior is one of the most important issues in game theory. Previous studies have shown that cooperation can evolve only under highly limited conditions, and various modifications have been introduced to games to explain the evolution of cooperation. Recently, a utility function basic to game theory was shown to be dependent on current wealth as a conditional (state) variable in a dynamic version of utility theory. Here, we introduce this dynamic utility function to several games. Under certain conditions, poor players exhibit cooperative behavior in two types of chicken games (the hawk-dove game and the snowdrift game) but not in the prisoner’s dilemma game and the stag hunt game. This result indicates that cooperation can be exhibited by the poor in some chicken games. Thus, the evolution of cooperation may not be as limited as has been suggested in previous studies.
Collapse
Affiliation(s)
- Hiromu Ito
- Department of International Health, Institute of Tropical Medicine, Nagasaki University, Nagasaki, 852-8523, Japan.,Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 432-8561, Japan
| | - Yuki Katsumata
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan
| | - Eisuke Hasegawa
- Laboratory of Animal Ecology, Department of Ecology and Systematics, Graduate School of Agriculture, Hokkaido University, Sapporo 060-8589, Japan
| | - Jin Yoshimura
- Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 432-8561, Japan.,Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan.,Department of Environmental and Forest Biology, State University of New York College of Environmental Science and Forestry, Syracuse, NY 13210, USA.,Marine Biosystems Research Center, Chiba University, Uchiura, Kamogawa, Chiba 299-5502, Japan
| |
Collapse
|
6
|
Ito H, Katsumata Y, Hasegawa E, Yoshimura J. What Is True Halving in the Payoff Matrix of Game Theory? PLoS One 2016; 11:e0159670. [PMID: 27487194 PMCID: PMC4972255 DOI: 10.1371/journal.pone.0159670] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2016] [Accepted: 07/05/2016] [Indexed: 11/19/2022] Open
Abstract
In game theory, there are two social interpretations of rewards (payoffs) for decision-making strategies: (1) the interpretation based on the utility criterion derived from expected utility theory and (2) the interpretation based on the quantitative criterion (amount of gain) derived from validity in the empirical context. A dynamic decision theory has recently been developed in which dynamic utility is a conditional (state) variable that is a function of the current wealth of a decision maker. We applied dynamic utility to the equal division in dove-dove contests in the hawk-dove game. Our results indicate that under the utility criterion, the half-share of utility becomes proportional to a player's current wealth. Our results are consistent with studies of the sense of fairness in animals, which indicate that the quantitative criterion has greater validity than the utility criterion. We also find that traditional analyses of repeated games must be reevaluated.
Collapse
Affiliation(s)
- Hiromu Ito
- Department of International Health, Institute of Tropical Medicine, Nagasaki University, Nagasaki, Japan
- Graduate School of Science and Technology, Shizuoka University, Hamamatsu, Shizuoka, Japan
| | - Yuki Katsumata
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, Shizuoka, Japan
| | - Eisuke Hasegawa
- Laboratory of Animal Ecology, Department of Ecology and Systematics, Graduate School of Agriculture, Hokkaido University, Sapporo, Hokkaido, Japan
| | - Jin Yoshimura
- Graduate School of Science and Technology, Shizuoka University, Hamamatsu, Shizuoka, Japan
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, Shizuoka, Japan
- Department of Environmental and Forest Biology, State University of New York College of Environmental Science and Forestry, Syracuse, New York, United States of America
- Marine Biosystems Research Center, Chiba University, Uchiura, Kamogawa, Chiba, Japan
- * E-mail:
| |
Collapse
|
7
|
Frame AM, Mills AF. Condition-dependent mate choice: A stochastic dynamic programming approach. Theor Popul Biol 2014; 96:1-7. [PMID: 24996205 DOI: 10.1016/j.tpb.2014.06.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2013] [Revised: 06/16/2014] [Accepted: 06/19/2014] [Indexed: 10/25/2022]
Abstract
We study how changing female condition during the mating season and condition-dependent search costs impact female mate choice, and what strategies a female could employ in choosing mates to maximize her own fitness. We address this problem via a stochastic dynamic programming model of mate choice. In the model, a female encounters males sequentially and must choose whether to mate or continue searching. As the female searches, her own condition changes stochastically, and she incurs condition-dependent search costs. The female attempts to maximize the quality of the offspring, which is a function of the female's condition at mating and the quality of the male with whom she mates. The mating strategy that maximizes the female's net expected reward is a quality threshold. We compare the optimal policy with other well-known mate choice strategies, and we use simulations to examine how well the optimal policy fares under imperfect information.
Collapse
Affiliation(s)
- Alicia M Frame
- Department of Biology, University of North Carolina, Chapel Hill, United States.
| | - Alex F Mills
- Department of Operations and Decision Technologies, Indiana University, Kelley School of Business, United States.
| |
Collapse
|
8
|
Yoshimura J, Ito H, Miller DG, Tainaka KI. Dynamic decision-making in uncertain environments II. Allais paradox in human behavior. J ETHOL 2013. [DOI: 10.1007/s10164-013-0359-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
|