Archive for November, 2016

Polar Bear & Seal Dilemma in the Apollonius Circle


Polar bears pursue seals. But polar bears cannot swim as well as seals. As the ice cap melts, there is less ice caps for the seals to rest. Both have to survive. With melting ice caps, the seal as victim should adapt to the same speed as the pursuer, the polar bear. In the language of differential games equal speed preserves the distance in the Apollonius circle Swimming at equal speed means survival. A valuable lesson – more cooperation, less conflict.


One of the classic illustrations in the game theory literature is the Prisoners’ dilemma. Faced with the option of whether to cooperate or compete with a rival depends on what each believes the other is going to do. In the playbook of a rational player competing is a dominant strategy supporting a Nash equilibrium wherein both are worse off in the long term.  Is there an equilibrium we all prefer? In nature, there are many games and different equilibria: the polar bears hunt seals; the seals swim to avoid the bears: This ice game of pursuit results in a Darwinian outcome of survival of the fittest.

Does it have to be so? Can the rules of the game be changed? What if the polar bears and seals signal mutual cooperation in order to survive? They cooperate because their behaviour is directed by the fact that the ice is melting. The amount of ice is no longer infinite. The newly observed patterns of behaviour in nature are too complicated to understand. New insights are being discovered in the BBC’s Planet Earth So, what if cooperating is a signalling solution to the polar bear and seal dilemma, how can we rationalise this solution from a game theory perspective? And how can we observe a degree of cooperation?


As the ice cap melts, there is less ice caps for the seals to rest. Less resting places makes it harder for the polar bears to capture the seals as they spend more time swimming in the waters. Polar bears cannot swim as well as seals. So, do polar bears spend more time on fewer ice caps waiting for the seals to rest or do they improve their ability to swim longer in pursuit of the bears?

For the seals, knowing that polar bears cannot swim as well and as long as they can, the optimal action is to stay longer in the water. But they risk capture by the polar bear, who, knowing that the seals believe that polar bears cannot swim as well as they, adapts to swim better, pursues the seals until they rest on an ice cap in the belief that there will be no polar bears nearby or the seals believe that the bears cannot swim as well as seals.

 Thief of Nature

In a post-Darwinian world of melting ice caps more seals stay longer in the icy water. If more seals stay longer in the icy water polar bears unable to swim as well as seals, will fall in population as they lose a vital food source. Fewer seals will survive. Fewer seals means fewer polar bears, so as the ice caps continue to melt it appears to be a no-win outcome for both the polar bears and the seals. This ‘thief of nature’ no-win outcome obtains since neither are responsible for the ice cap melting.

So, is there a lesson for the seals and bears faced with the dilemma of survival? There is the Darwinian outcome of survival of the fittest. However, since climate shapes their environment[1]  whomever adapts better to the melting ice caps has a greater probability of surviving. But what if they signalled to each other – the predator bear signalling to the seal?

Avoiding Bad Choices

Fish congregate in the centre of the frozen ice pack where the ice is thinnest. You, the reader and I, as humans, we know that if either of us falls through the frozen lakes of Siberia in temperatures approaching 60 degrees below zero we will die of hypothermia, or if you rescue me, my body will freeze immediately and I will die of shock. Same fate applies to you, I’m afraid. It’s nature’s own Siberian dilemma. We both know that this outcome of very bad choices to be true.

A higher payoff is represented by avoiding thin ice, by avoiding bad choices. The seals probably know by now that both will obtain the higher payoff if the polar bears stop pursuing them. And the polar bears believe that the seals believe this to be true.  Neither benefit from thin ice and both need the ice pans to survive. So, the polar bears unable to swim as well as the seals adapt to new food sources. The bears signal to the seals and both survive by adapting in the game of melting ice caps. A signalling strategy represents a payoff-dominant Nash equilibrium for both. By surviving they could both reach a payoff-dominant Nash equilibrium.

Swim at Equal Speed

In order to understand this signalling strategy we need to transpose the dilemma into a pure search game. A solution is for the seal as victim to adapt to the same speed as the pursuer, the polar bear: equal speeds equals survival in the equation of climate change. Same equal speed minimises capture provided each follow a pure strategy of avoiding thin ice. In the language of differential games equal speed preserves the distance in the Apollonius circle. The animals may rely on chance to choose their paths to survival. Seals can swim; polar bears need to learn to swim as fast as seals. The sole decision rests with the seal, the victim in the search game: how fast should she swim? Not swimming is not an option. Swimming too often or too fast is tiring and with the increase in thin ice, resting ice packs are less promising.


The life history of seals and polar bears is a cumulative process of adaptation of means to survival that change as the ice caps melt. What is possible is whatever isn’t necessarily not the case[2], so possibly, seals and bears swim at equal speed. Logically, that means that there is at least one possible world in which the sentence ‘seals and bears swim at equal speed’ is true. Albeit, since you and I are responsible for the melting ice cap, we too, need to adapt. We need to avoid conflict. There is at least one possible world in which the sentence ‘to avoid conflict’ is true. There may be and it is our 21st century world with finite resources and infinite data.

[1] On polar bears: and on seals


[2] Excellent read is Daniel Dennett (2003): Freedom Evolves

Game Ontology

Patrick will be on a speaking tour in Asia introducing an innovative approach to strategy by complementing game theory with philosophy. The phrase ‘game ontology’ is reserved as a working title for a new book planned with McGraw Hill Asia for 2017 release. He will introduce the philosophy of conduct to audiences at business meetings in Hong Kong on March 17th and Singapore March 22nd. The presentation in Hong Kong will introduce a theme of apoptosis to data patterns, wherein individual choice can be eliminated by a programmed sequence of events. The Masterclass in Singapore will more formally introduce the game ontology underpinning a cognitive business strategy framework and blend the episteme of a mistake into the thinking behind a strategy plan. The big idea behind cognitive business strategy is to provide senior management in an era of big data with an Aide Memoire on how to think ‘ahead of the mistake’ in the translation of a value proposition into a winning strategy. The attached papers are preliminary and subject to change and comments are welcome.


Hong Kong: Flyer.pdf
Singapore: cognitive business strategy 22mar2016.pdf