Across the world we are adjusting and adapting to life under lockdown, furloughed, cocooning, and moving around within an agreed social distance. We are responding to a new world order under Covid, a new normal with a greater degree of mutual interdependence as people observe and respond to each other’s behaviour. While there are computer games[1] to teach the importance of social distancing, we wondered whether or not game theory reasoning could underpin our behaviour during the Covid pandemic. The essay is about game theory reasoning and its putative relevance to rational behaviour under Covid. Related research into the aesthetics of game theory reasoning in literature inspired a reading of Sophocles’ Antigone and Beckett’s Waiting for Godot and Watt. The research[2] into obedience and waiting is ongoing but some thoughts are shared here as we design a game of obedience and waiting with a view to understanding of our own rational behaviour during this Covid pandemic. In these unsettling times, if one can only think of something as unbelievable as invisible Zen archers then there is hope in knowing that we are all facing this together.
Obedience as a Waiting Game
Sophocles’ play Antigone provides an historical canvas. The play is about pride and responsibility but, it is also a story about the rule of law, the role of Government and civil disobedience. Antigone is also[3] a ‘public story’, in terms of obeying a universal law. Should Antigone have adhered to the rule of law, the law of the State, and thus, obey her uncle King Creon? Or, choose not to obey. Like Antigone we have free will. Like Vladimir and Estragon we wait. So, do we adhere to the rules and obey Government guidelines, or do we disobey? The same reasoning would apply to groups and (economic) clubs[4] wherein members adhere to rules and regulations. Rational players as individuals do commit to an organised set of rules.
It has been shown by Schelling[5] that it is ‘not always safe to assume (individual) outcomes accurately express the individual decisions’ as observed. Therefore in the search for rationality in a game of obedience, there may be a sub-game ‘trembling hand’ equilibrium in waiting, if we can interpret waiting as obedience. Our research has focused on an optimal design of a game of waiting inspired by a reading of Beckett’s Waiting for Godot and Watt. And suffice to note here in our design that the etymology of ‘waiting’ is either an interpretation of ‘hoping’ [Latin verb ‘spero’] as in Antigone’s behaviour, hoping to bury her dead brother or an interpretation ‘remaining in place’ [Latin verb ‘manere’] like Vladimir and Estragon as they wait for Godot. The latter interpretation is applied in this Blog essay. That is, we either obey Government restrictions and wait for further guidance or we disobey.
Ontological Insecurity
But there is an added poignancy to Sophocles’ narrative today during the Covid-19 pandemic as citizens in particular begin to question and debate Government policy. Covid-19 has disrupted individual and collective confidence. It has triggered what sociologists[6] and anthropologists[7] call ‘ontological insecurity’- an anxiety about your safety and security at a moment in time. We are now in a state of perpetual angst. It can be best understood by your answer to the question: how safe do you feel? We need assurance: in the medium term the roll-out of a track and tracing app or the wearing of masks might suffice. Masks or face coverings, for example, provide confidence and security: you wear a mask to protect others and they wear a mask to protect you from Covid infection. A vaccine would provide long term assurance. However, while it is sufficient to observe others taking precautions it is necessary for our own security that others, have confidence in their own survival. Therein lies the mutual interdependence that is characteristic of game theory.
Player Unknown
Like Antigone, we as citizens are now players in a game of obedience. The game is between you, Player 1, and ‘others’, referred to as Player Unknown (PU).[8] The PU allows us to blend the ontological insecurity of rational players into the game design. Antigone did not have to obey her uncle King Creon. She chose not to. Ultimately, like Antigone your playbook is a question of individual choice versus fate. There are many scenarios? What if King Creon did not expect Antigone to bury her brother Polynices? What if Estragon followed Pozzo and opted not to wait for Godot? We opted not to explore these hidden games with mixed strategies in this essay. They are to follow.
We ascribe Covid-19 behaviour as obedience and in an obedience game we have illustrated the payoff to ‘others’ in italics in Table 1 and the payoffs to you are illustrated in bold. Every player is unknown to you and you have zero information on any player. PU provides a cognitive framework that enables you, as a rational player, to anchor your confidence to observed behaviour by any PU. How would you choose your strategy of play in such a game? Players are rational and optimise their choices to get the most preferred outcome. PU would like to avoid the game or transfer responsibility to others. The sub-game payoffs reflect this preference ordering. Although the payoffs in Table 1 are arbitrary cardinal numbers, they do reflect the players’ preferences in a game of obedience. As citizens we have become players in a Covid obedience game. The ethical trade-off, arguably, do X and not Y is reminiscent of the narratives in Sophocles and Beckett. We opt to frame the trade-off as a non-cooperative game.
Trembling Hand Equilibrium
The trade-off is between individual choice and fate. We wait, we obey. Or not. It is a strategic choice with a sub-game perfect equilibrium. In other words,[9] obey as a strategy, is the Nash strategy in the ledger of choice for the duration of the game. Obedience, may well be the sub-game node to the main game of Covid survival over pride. The player’s payoffs take the cardinal form (-1, 0, 1, 2). The equilibrium payoff (2,1), for example, can best be understood by the philosophy[10] of Montaigne: ‘my life has been destroyed by many catastrophes most of which never happened’.
Table 1: Sub-game equilibrium with Player Unknown
| Disobey
|
Obey | |
| Disobey | 0, 1 | 0, 1 |
| Obey | 1, -1 | 2, 1 |
To obey the rule of law, to obey the Government until lockdown is eased and a vaccine available presents each and every one of us with a rational choice to obey. That choice is best understood by adapting the words of an old Biblical tenet[11] sic ‘I obey because it is absurd’. Playing this game with rational obedience in the playbook will obtain the payoff 2 for you as the rational player. Disobedience is not an option for you. There is no later. Every form of behaviour is shaped by trial and error. Disobedience, however, may obtain an elusive payoff of 1 for PU with an inherent risk of -1. For Antigone, it meant her death. What if she no longer believed in her strategy of disobedience?
Topology of a Playing Strategy
So let us consider two equilibria: the (2,1) payoff and a ‘hidden’ equilibrium[12]. The equilibria are separated by a probability (function) of reciprocal altruism[13]. The selfish outcome is present in both but due to a payoff equation (yet to be defined) it manifests itself differently in the two equilibria. Therefore the altruism function could be a discontinuity that can be developed from the differential form of the equation of life[14].
Definition: An equilibrium is hidden if it does not intersect with a well-defined open neighbourhood of equilibrium points. Or it may be a point in a small open neighbourhood whose closure is compact.
Theorem (to be proven): The (2,1) equilibrium payoff is a subset of a larger group of non-observant ‘hidden’ equilibria corresponding to a topological vector space wherein a convergence is defined in the neighborhood of saddle points.
Corollaries: I: The neighborhood of saddle points include a neighbourhood of 0. II: The vector space would include the payoffs at the default position of selfish behaviour by PU and the payoffs with co-evolution of the players who ascribe to a reciprocal altruism type.
Prognosis
Until further research is concluded, the payoff (2,1) is a trembling hand equilibrium. It can only occur if PU commits to obey as a strategy. So, ask yourself: should you obey? Affirmatively, yes. In a Bayesian game, Nature reports to each player his or her type. Are you Player Unknown? The equation of life in a Covid game is analogous to a Krepski two-move game of chess at a moment in time: opening move is either obey or disobey. If you disobey there is a greater risk of death. Checkmate. But this checkmate can only occur if one player (say, White in chess) commits to a mistake[15] so it seldom happens in practice. An anonymous rational PU in a two-move game does not want to die anonymously. If you obey the rules and guidance, you could evolve as a spectator to obedience, secure in the knowledge of your own survival and the survival of others. Despite disobedience by PU, behaviour will tend to cluster around (2,1) because there are many more rational players like you subject to the Gaussian distribution of rational behaviour. There is hope. The equilibrium point of balance in this stylised obedience game of waiting is realised when you can begin[16] to construct ceremonies out of the air and breathe upon them’. The Zen archers become visible. Keep safe. Keep sane.
[1] Check the Thomas Reuters Foundation article by Emma Batha, ‘Covid-19 Computer Games Reaches Children Importance of Social Distancing’, 19 May 2020
[2] Collaboration with Manfred Holler at LMU in Munich ‘waiting games’ inspired by the literature of Beckett and Brecht and the narratives in both Greek tragedy and Roman comedies, in Sophocles and Seneca respectively. The idea is a Beckett player type delaying and thinking [like the ‘sanyasin’ in Hindu] as observationally equivalent to waiting. Interesting perspective from Kimberley Bohman-Kalaja who published Playing the Spectator While Waiting for Godot at Princeton University Press, 2007.
[3] The view expressed originally by Holderlin who translated the play from Greek into German.
[4] McNutt, Patrick (2002): Economics of Public Choice Elgar Publishing, UK.
[5] Schelling’s Game Theory p241 by Robert Dodge (2012) Oxford University Press.
[6] Framed by Giddens (1990) in his book Modernity and Self Identity as ‘ontological security’ with the earlier original concept of insecurity attributable to R.D. Laing (1960): The Divided Self Penguin, London.
[7] Refer to the R.D.Laing’s research overlap with the grid-group analysis of Mary Douglas and Margaret Mead and application in an earlier article McNutt (1987): ‘Anthropology, Economics and the Socio-Economy’ The Arab Journal of the Social Sciences vol 2 no 1 April.
[8] Adapting the strategy from the gaming fraternity’s Player Unknown Battle Ground (PUBG): worksheets from my lecture notes ascribed to the MBA cohorts who attended Masterclass at both Manchester and Smurfit Business Schools in Manchester and Dublin.
[9] Check out McNutt (2010): Decoding Strategy published by McGrawHill and Holler (2018): The Economics of the Good, the Bad and the Ugly, Routledge.
[10] Selected from Montaigne’s Essais (1595) as translated in the essays in Wikisource.
[11] Adapted from an old Christian tenet ‘credo quia absurdum’ – ‘I believe because it is absurd.’
[12] The term suggested by colleague Manfred Holler at LMU and Hamburg.
[13] From McNutt (1988): ‘A Note on Altruism’ International Journal of Social Economics vol 15 pp62-64.
[14] The equation of life to be a diffusion like equation expressed in terms of the movement from selfish to altruistic behaviour in a Covid-type obedience game.
[15] The game design with a mixed strategy would assign probabilities to a second move for Antigone. What if she no longer believed in her strategy of challenging her uncle?
[16] Adapted from the narrative in Cormac McCarthy’s 2006 novel The Road at page 78 ‘where you have nothing else construct ceremonies out of the air and breathe upon them’.