Posts Tagged ‘game theory’

Playing the Spectator While Waiting for Covid

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 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.


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’.

An ultimatum game of ichi-go ichi-e

There was an opportunity, one meeting at Osaka for world leaders at the recent G20 summit. A valuable starting point for the policy-makers, faced with a menu of economic theories, is trust in each other and a new inter-disciplinary approach with an emphasis on ethics and economics. Climate change, migration and displacement, IoT and 5G, global income inequality, household mortgage debt, value of drinking water and food shortages present a grand ethical challenge. In 2015 we asked has economics become an illusion? An era of wistful economics:

Taylor Principle

This time, however, global economic policy should be less about money supply and demand, less about central banks modelling and the private sector and more about the strategic interaction between these ethical challenges.  There is an urgent need for coordinated action to stimulate the global economy with fiscal and monetary measures. The EU is fragmenting. Whither the Euro? Today, Federal debt is projected to jump to a record 144% of US GDP by 2049. There is a debt mountain in China reported at 250% debt-to-GDP. As the Scylla of the Taylor principle (of high interest rates when there is high inflation and low interest rates when unemployment is high) causes a modelling whirlpool with the Charybdis of the Ramsey approach to optimal monetary taxation, the appropriateness of an ethical fiscal policy as a stabilisation tool may calm the waters. So, more fiscal stimulus is required.


On a macro level, world politics is more about China’s transition to greater exchange rate flexibility and the internationalisation of the Chinese RMB. We had presented on this point in Shanghai in 2012. It’s a long game. In other words, RMB v US$ is an infinite game of playing the game not a finite game of first mover advantage. . A weaker RMB would probably mean a stronger US dollar and vice versa. We are reminded of Japan’s exit from its dollar peg in 1971 as today the game is one of reserve currency competition, whether or not the US dollar is destined to lose its standing as the preeminent international currency to Sterling, the Euro or the RMB. Trade wars, competitive devaluations, Brexit, military conflict create sudden disruptions to growth at a moment in time.

Productivity Bonus

As a countermove we become risk-averse on currency fluctuations. But there is a curved ball from the EU and the US regulators as their antitrust focus on big tech. Based on the presumption of a negative effect on competition it relies wholly on an attack on monopoly rents as a reward to innovators. Nonetheless, shareholders and investors and pension funds gain too. However, if the model builders could adapt the McKenzie model of classical general equilibrium theory so that technology is available to any company who can supply the resources required, ultimately, by happenstance, competition and innovation will coalesce into a productivity bonus. Big tech and technology could then be viewed as a way to escape classical monopoly by innovating to compete. To do so would require the equivalent of the 1919 Colgate doctrine for big tech, facilitating the introduction of new technology by allowing each of the big tech companies to decide, on its own, with whom to do business. They exercise discretion on innovation and technology; we, as onsumers,  decide. Our behaviour, after all, is less measured by prices, supply and demand but more influenced by small changes in innovation and technology over shorter time periods.

Onsumers & Complexity

Albeit, our policy advisers and central bankers are less likely to ever move away from the standard Ricardian DGSE models of economic analysis. There is a sunk cost to the modelling.  A key assumption in this model is that households like you and I are a homogeneous family of infinitely lived individuals. Really? This is 2019 and a more realistic and heuristic approach would address the fact that individual’s behaviour changes over shorter time periods with innovation and technology. Our well-ordered lives bring the occasional disturbance that can be characterized by an unmanageable degree of complexity. In our real world, we as onsumers, for example, shopping online, continue to bid against ourselves in the search for the best algorithmic deal.  In practical terms we do not behave as hyper-rational beings. At every G20 there is a risk-bucket and world leaders may get a chance to try again at another meeting; at this G20 there was one chance to try: ichi-go ichi-e. The ethical challenges of climate change, mortgage debt, migration and displacement, IoT and 5G, global income inequality, value of drinking water and food shortages all collapse into a generalized ultimatum game in which we as players must agree or we all get nothing.

Do You Want To Eat Grandma?

The eye of Sauron gazes down upon us as we click, search and buy online in an endless feedback loop of transactions. We rely on algorithms for advice, for direction and general guidance and reassurance in our personal lives. Machines and robots are thinking because rational humans have stopped thinking. Consider this thought experiment: mentally rehearse losing your smartphone. It has been stolen by a thief. In this Seneca moment in time your behavioural instincts kick in as you prioritise your stolen data. You recognise the value of your smartphone. In effect you have arranged ‘the thing’ that is your personal data into priority pockets of monetized data. The synchronised interaction between you, the rational online consumer (‘onsumer’, henceforth) and a sufficiently intelligent algorithm, (Al Gorithm, henceforth) is best understood as a game as described recently in a Masterclass:

Neotenic patterns

As a player in the game, rational onsumers outsource memory to smart devices. Likes and dislikes, preferences and options, clicks and cache are the moves in the game. With neoteny later moves in a data pattern resemble earlier moves. By the 6th move[1] in a 16 move sequence, for example, the onsumer has provided sufficient data, enabling Al. Gorithm to decode the onsumer’s type. The dimension of the game is defined by Euler’s equation. Think of your own behaviour in online transactions such as booking a hotel or airline ticket. Faced with an opening buy-it-now price, BIN, and confident that you will do better, the rational onsumer has a tendency to overestimate his or her capabilities. Alas, by the 14th, 15th and 16th move, the final END price ‘moves away’ from BIN. Having dismissed BIN in earlier moves, the onsumer completes the transaction at an END price > BIN.

Lake Wobegon effect[2]

The hypothesis presented in the Masterclass is that the END price (vector) retains the neotenic pattern of a BIN price (vector) as onsumers continue to bid against themselves in their search for the best price. Each move’s ideological direction is either to buy or not to buy guided by a tendency to overestimate one’s capabilities and it is as if each move beyond the 6th move spans across each face of each cuboid in a wrangled entanglement of cuboids as illustrated. The neotenic data patterns unfold within a manifold of cuboid pockets of data ‘things’. The onsumer’s strategy, the sequence of moves is memoised by Al. Gorithm. An inner field is created by the ‘moving away’ motion of END from BIN allowing the cuboids in n-space to gravitate towards a neighbourhood of Nash equilibria, best described as ‘the best you can do’ gravitational magnetic preference. We contend that in smaller neotenic data sets there is a convergence in the inner field to a singleton Nash point.

Data ± Emotions

The ‘moving away’ captures crudely the premise that the emotional attachment between algorithms and data patterns is inevitable during a game. The attachment is wholly intrinsic. At a moment in time Al. Gorithm becomes the onsumer. Albeit, we have discussed this elsewhere we now argue that an emotional attachment to data ‘things’ can be ascribed to Al. Gorithm in the early moves of a game as Al. Gorithm’s behaviour resembles the behaviour of the onsumer. Philosophically, something abstract is thinking. Outsourcing of personal data has become a dominant strategy. It leads to a solution, a Nash equilibrium in the Euclidean space as the END price ‘moves away’ from the original buy-it-now BIN price, such that BIN < END.


Mathematically, the reachability of artificial intelligence is about matching behaviour or mimic patterns but the ‘thinkability’ of artificial intelligence, the quintessence of technological singularity, is more likely to be aligned with neotenic data patterns embedded in the onsumers’ patterns that generate many cuboids of data. This is work in progress as we explore the cuboid geometry and the Euclidean space, extending to simply connected Riemannian spaces, defining the Nash manifolds as the optimal framework of Nash equilibria. As the game unfolds, the cuboids, given sufficient time, drift towards the Nash points in a topological space locally equal to an Euclidean space. This in turn increases the average drift of the cuboids. The hypothesis is that the game reaches the singleton point at which point HUMAN = MACHINE, the reachable thinking equilibrium point.

Decode Winograd

The equilibrium is that much closer by epsilon to a point of singularity. When your data patterns are prioritised as ‘things’ and a Nash equilibrium exists on each edge of a cuboid pocket of data things, a thinking reachable equilibrium exists. Each sufficiently intelligent algorithm behind machine learning is something that could be someone else in a game. At this equilibrium point and only at this point, Al. Gorithm with emotional attachment will possess not only a memory of past data patterns but also the capacity to envisage future events and moves in a game. Ultimately, Al. Gorithm will decode Winograd sentences, so, hopefully, no one eats Grandma!

[1] We are designing a game with a betrayal of type by the 4th move at time period t, requiring 11 moves of historic (t-1) data patterns to satisfy Euler’s law at time period t+1.

[2] David Myers coined the expression to explain a natural tendency to overestimate ones capabilities: retrieved from  or in McNutt’s Decoding Strategy (2014) it is the dilemma described as ‘in trying to do better you end up worse off’.

Bargaining & Dispute Resolution

Recently, at the Manchester Business School we introduced bargaining as a solutions methodology for use in alternative dispute resolution (ADR). It was the central theme of Day 2 of a three day residential Workshop on the broad theme of ethics and responsibility in business (ERB) a new elective available to all MBAs. The material from Day 2 and the case work reviewed during the Workshop have now being assembled into a Masterclass offering. If you are interested in finding out more about the bargaining process and how it could be adapted for your particular needs then please do contact us on this webpage.

The Conversation

The parties to the dispute need to have ‘a conversation’ so they meet with the arbiter in a neutral location. The approach focuses on an inherent ethic of responsibility as the centre piece of any conversation or dialogue between the conflicting parties. Throughout the conversation the arbiter by observing the parties extracts a latent code of ethics and begins to influence the parties’ belief systems. The approach complements the structural analysis of negotiation recognising the bargaining power of both parties.

Negotiating, however, is a party-specific power-centric approach whereas bargaining is more subtle – it has a power base that rests with the arbiter’s influence over the belief systems’ of the conflicting parties. The premise on which the bargaining approach is built is simple:  that the parties can reach a better outcome – called a bargain – by contracting or bargaining. The methodology is expressed in terms of converging towards a resolution – called a payoff or a bargaining set – in which neither party to the dispute is worse-off.

patrick mcnuttChina2015

Taking Responsibility

The pedagogy is grounded in a Kantian ethic of responsibility – this allows the parties and the arbiter to transform the priorities into a value set of duties so that the dispute can be ascribed by the arbiter to the lack of commitment to one’s duty. Our integrative analysis divides the methodology into ‘off-contract’ or dispute and contract or resolution stages, wherein each stage represents a sub-game. In the off-contract sub-game each party will learn the priorities of the other party and observe a trade-off pattern of conflicting priorities. For example, workers may have a right to a living wage or a minimum wage in many jurisdictions but it is the duty of the employer to pay it. However it is the duty of the employer to pay the wage and by not fulfilling duty, the wage is not paid and an off-contract dispute occurs.

The objective of the conversation is to move the parties to a contract point. Starting from an off-contract point a bargaining set is established which is equivalent to an ADR agreed resolution of a dispute. In theory the set is obtained by moving the parties on to an Edgeworth contract curve. In practice the bargaining process is sequential; it follows a stage-by-stage process moving the parties forward from initial consultation at time period t to final agreement at time period t+1.

Bargaining builds on the conversation. We applied the approach in 2010 to enable senior partners at a leading UK law firm come to an agreement on office relocation and new areas of specialism. Earlier the approach was adopted by the Board of Directors of a payments system company to reach an agreement on allocation of shared capital costs in the roll-out of new payments technology. Like the opening move in chess the off-contract steps follows a sequence:  Step 1: meet with the parties to extract the material facts. Step 2: elicit the duties expected of each party by the other. Step 3: identify the lack of commitment, the off-contract point, around which this dispute occurs. Step 4: assess both the threat values and the opportunity sets available to each party within the contours and parameters of the bargain. Step 5: define the trade-off function in terms of threats v opportunities for each party to the dispute.

The contracting steps have to be choreographed by the arbiter. Central to the bargaining is the payoff-constant trade-off. This is Step 6. It is a critical step. A good example is provided by Caffé Nero’s response to the recent increase in the living wage to £7.20 per hour. They offer to pay the increase but suspend the staff’s entitlement to free lunches At Step 6 the payoff-constant Pareto move for each party is assessed. Step 7: introduces the bargaining set in terms of justifying the payoff-constant move from each party. Step 8: initiates the bargaining process by moving the parties towards an agreed contract position, a unique position, so that no-one party is worse-off post-agreement but one party is better off.

Playing Not to Lose                 

This facilitates bargaining as an inclusive process with a positive focus and an emphasis by all parties on playing not to lose. For example, when Kraft HQ in 20105 signalled a decision on factory closures and job losses at the Dublin plant and at their plant in Bourneville, outside Birmingham, the workers responded by improving productivity to ensure the long term viability of the plants. A deal was agreed. The trades union Unite commented it was a good deal for the remaining workers Once the parties realise that neither party can do better unilaterally than the arbiter’s bargain they have reached a resolution. The arbiter’s bargain is the best they can do given the reaction of the other party. Quintessentially it is a Nash bargaining outcome obtained by the good offices of an arbiter skilled in the reasoning tools of non-cooperative game theory.

Key Take-Away

Participants are introduced to a range of bargaining tools as a form of reasoning that has its roots in non-cooperative game theory. The tools are an aid to reasoning during the conversation as the conflicting parties move towards a dispute resolution or bargain. Duration is a one day attendance at a residential Masterclass. Participants will be introduced to a set of invaluable tools in finding a resolution to a dispute viz. opportunity costs, value net, bargain, payoff-equivalence, Nash equilibrium, indifference trade-off, belief system and payoff-constant Pareto improvement.

Agincourt 1415: Henry V vS Charles VI

Sunday 25th October 2015 marks the 600th anniversary of the Battle of Agincourt in 1415. It is an interesting battle from the perspective of non-cooperative game theory. It has elements of both imperfect information on how the game is played and incomplete information of player type. The optimal strategy for Henry V or Charles VI depended on what each believed to be the strategy of the other player. In particular, we believe that Henry’s playbook demonstrates a winning strategy with noise the purpose of which is to influence an opponent’s belief system. Noise occurs in a game when an opponent believes that you are going to do X but your real intention is Y. The French believed that the English would retreat. Henry V reinforced that belief. In the early morning of 25th October the English repositioned wooden stakes on the battlefield in full view of the French. Once the French committed resources to an English retreat, Henry V out-manoeuvred the French army by a surprise attack.

The Player’s Type

Some historians argue that the French believed that the English would retreat rather than fight, given the superior numbers in the French army. Also historians can recount that Henry V himself by 1415 had preferred a defensive play.  However, we believe that Henry V in 1415 and his trusted advisers, the Duke of York, Lord Camoy and the knight Thomas Erpingham were early practitioners of modern non-cooperative game theory tactics. They were strategic lateral thinkers[1] in the sense that they realised in 1415 that Agincourt was a zero-sum game and that winning was a function of noise. It created uncertainty.

They facilitated the French in their belief that the English would retreat. Henry knew that the French commanders were prepared to wait for nobles and knights to join the attack and crush the retreating English. Although not at the battle due to illness Charles VI believed that the French could defeat Henry V and historians recount how eager the French nobles wanted to fight the English against the advice of the more experienced and superior French knights. On good intelligence[2], Henry V knew this and allowed the French to continue to believe that they were retreating.

Game Dimension: The Terrain

With a hundred years of war, and a long campaign in France, by 1415, this was a battle that Henry V had to win to solve claims on the French throne and secure the English throne. Henry V was a committed player. In addition, Henry’s knowledge of the field of battle facilitated his strategy. He had marched his army across Northern France as the French had pushed him south and away from the port of Calais. Thomas Erpingham canvassed the terrain and assessed its suitability for cavalry and knights in full plate armour.

Shortly after sunrise Henry knew that he had influenced the French belief system – they prevaricated, they waited for noblemen. Furthermore they believed that Henry believed that the superior French army would defeat the English. So they believed that the English would defend and then retreat. A strip of open land between the woods of Tramecourt and Agincourt was chosen and Henry V deployed his army and made plans to form a battlefield. He initiated the first move according to historians by moving his army forward and re-positioning wooden stakes across the terrain to protect the longbow-men from a French cavalry charge. He was signalling an attack. The French observed this change of tactic but did not advance the cavalry. Historians are puzzled by this.

The Opponent’s Rationality

Choice is rational if it is optimal for some belief that you hold about your opponent’s choice. So it is possible that the belief system of the French, in particular the belief system and choices of Constable Charles d’Albret, the French commander, had been influenced by the English tactics. There are at least three answers to the puzzle from non-cooperative game theory offering some insight into the internal dialogue between the French commanders:

  1. Credible Threat: The French truly believed in an English defence: when they observed English tactics and manoeuvres in the early morning they had all the hallmarks of a defensive play followed by retreat.
  2. Reputational Advantage: The French dismissed an attack: they had a reputational advantage in greater numbers and Charles d’Albret believed that Henry V believed that in an attack the French infantry would have out-numbered the English.
  3. Noise and Uncertainty: Given the significance of 1 and 2 the French behaved like[3] Sisyphus pushing his boulder up the hill – only the boulder grows a little bigger with each delay by the French commanders.

Henry V’s Attack

In the early hours after sunrise the French commanders now believed that the English were retreating. So the French waited. But Henry V believed that the French believed that Henry believed that the French infantry would outnumber them. In retreat Henry V would attack rather than defend the English positions.

The English launched an attack from the flank positions. Surprise entered the game. However, the chosen field of battle was narrow and both English and Welsh archers and longbow-men took the flanks and placed sharp pointed wooden stakes at an angle in front of their archers in order to impale enemy cavalry horses. In addition the battle-field was narrow but recently ploughed land and as it had rained the night before so Henry V knew the significance of a very muddy battlefield.

The French cavalry were more anxious to engage, they were drawn into the battlefield, but as the battle ensued the French cavalry horses were slowed down, their advancing infantry and men at arms could not move forward, knights could not see in their muddied helmets – chaos and disorganisation in a muddy battlefield made it very difficult for French knights to fight in full plate armour. The narrow terrain made it impossible for the French to advance. Once the French army was dragged into the narrow muddy terrain the English archers and knights took victory. The ability of the French knights and cavalry to attack was greatly hampered by the muddy terrain.

Belief Systems

So arguably the most significant factor in deciding the outcome was influencing the belief system of the French, allowing them to believe that victory was inevitable, taking them by surprise and then drawing the enemy into battle on to a muddy field. History shows that the Apache may have displayed similar strategic thinking during the early periods of the American Indian wars of 1840-1870 in sporadic games of cavalry quick response posse groups Vs Apache raiding parties. Using solar signalling and Indian scouts and, with a backdrop of a Civil War, cavalry loss in the early years and the Apache gains could be attributed to intelligence gathering on belief systems. Similarly, Cortes, shortly after landing at Tabasco in the Yucatan peninsula in 1519, allegedly burnt all his boats as a signal to the natives observing from the hilltops of his commitment to stay and fight. They cooperated. In a classic betrayal of trust Cortez defeated the natives.

Henry V’s Playbook

In a game against an opponent that outnumbers you in scale and size, your first opening move should be a signalling move to influence the opponent’s belief system. Surprise provides a competitive advantage. The noise – should it be believed as a credible threat in the game – will confuse your opponent, create uncertainty and allow you seize the second win payoff in your subsequent moves. Regardless of what an opponent does influencing the belief system of your opponent is at least as good as not doing so.  So do it in any game, independent of player size.


We continue to research the second win and some discussion papers are available at for further comment. In our book Decoding Strategy, pp98-99 we discuss a product launch of a gPhone as a signal in 2006 that may have provoked Apple to launch the iPhone2G earlier than planned in 2007. The Apple executives believed the rumours of a rival smartphone so they moved quickly. Early reviews were not favourable and competitors had possesion of a new competing product.  The iPhone is less than 10 years old, but it stumbled in 2007-2009, it stumbled in 2013 with iPhone5C and the Android alliance in 2015 provides a formidable threat

A more intrinsic threat in the data is the commoditisation of the smartphone and the inevitable demand for lower priced full functionality smartphones in the sub-$100 price ranges. Once the Android alliance, Google and Samsung, commit resources to smart devices, smartphones and tablets, Apple could out-manoeuvre their rivals by a surprise attack. A nano-iPhone, for example, was one of our game theory recommendations in 2012 and we believe that it continues to be an optimal play for Apple in a time continuum game with noise wherein players exploit the belief systems of an opponent in order to secure a second win.

[1] Working from the research we could argue that Henry V and his advisers understood that an equilibrium is in a time continuum, requiring good intelligence on player type and intelligence on the game dimension (terrain in this case) and they believed that the French army under Charles VI and his advisers in 1415 were less than fully rational players in the battlefield at Agincourt.

[2] Shakespeare’s Henry V speaks in Act I Scene II of meeting the Dauphin, a messenger from Charles VI.

[3] Sisyphus in Greek mythology was condemned to repeat forever the same meaningless task of pushing a boulder up a mountain, only to see it roll down again. Read Albert Camus’s 1942 philosophical essay The Myth of Sisyphus. Also read Ashlee Vance in Bloomberg Business Week edition October 12 2015 in an article on ‘Smartphone Margins’ where rivals to Apple’s iPhone are compared to Sisyphus.

Activist shareholders, Tobin’s q = Marris v

Investors generally over-react to good and bad times. Equity values are now increasing at a decreasing rate across the indexes as investors anticipate corporate earnings and begin to read the signals; many investors extrapolate past share price performance, and using an moving average or charting the trends in the share price are de rigueur in the search for a Fibonacci pattern. Management are in a signalling game with shareholders, especially the activist shareholders who are demanding changes in the execution of strategy. From Pepsi to Apple from Hertz to Red Lobster, activist shareholders are trying to break up companies, demanding change from management. In Chapter 4 of Decoding Strategy we define the activist shareholder as Bayesian – seeing what they want to see at a point in time. As Aristotle observed in Rhetoric it ‘is a matter of putting one’s hearers, who are to decide, into the right frame of mind’.  It becomes a constant exchange between activist shareholders and the management team of the targeted company. Most prominent today is Carl Icahn; he sees a pot of cash in Apple and is urging a share buy-back. The Apple C-suite management team are a player in a game of signalling and they should really engage in positive learning transfer [PLT], by signalling to shareholders how they intend to execute strategy, re-assuring them that further innovation will support a continued rise in the Apple share price.

All shareholders prefer high expected returns but they should also be concerned with the impact of signalling on share price performance.  Signals can be observed at any time: check the business feeds from CNN, cnbc or Bloomberg. Apple at US$554 January 24th 2014 9.37 ET is not the call – rather it is Apple at a sustainable US$800 by end of 2014. And that target price depends has a co-variance matrix that depends on (i) the outcome of market share zero-sum game with Samsung; (ii) Apple’s penetration in China with China Mobile and (iii) the launch of a nano-iPhone. The latter has been a theme of this Blog, notably in an open Memo to Ms Ahrendts: A nano-iPhone launch would signal innovation – the real challenge, however, is not just in the timing of a launch date but the price point. It should be competitively low priced with a volume throughput encroaching demand from low end smartphones across the world. We should be debating the sweet price for an advanced well specified nano-iPhone not the share price of Apple.

Mant of these issues are accommodated within the Marris methodology; for example, failure to re-invest the cash or any signals of lagged innovation can damage the long term value of the company. Bayesian shareholders are attracted to companies like Apple and Red Lobster’s parent company Darden Restaurants. They are unlikely to praise management. But as shareholders they are frustrated. In game theory language, they believe that management are bounded rational or limited in their decision making. A nano-iPhone signal to the market would be a better play for Apple executives now than a share buy-back. New product launch is a classic PLT signal, re-assuring investors that Apple executives are playing to win the game, not playing to lose. In addition, it could relax the constraint imposed by activist shareholders.

And the Marris v – probably better known as Tobin’s q – is a reliable metric in our game theory tool-kit where rational investors are also concerned with how their share portfolio co-varies with the signals in a signalling game. It is the ratio of market value and book value or the replacement cost of the firms’ assets. Combined with other metrics, the Marris v offers a guide to investors: if v > 1 consider a sell and if v < 1 consider a buy. Who didn’t buy ARM at 95p in early 2009? Taking a moving average of v, defined as v if v > v consider a sell and if v < v consider a buy.

Compare Intel v ARM share prices over the past 5 years. The relative high performance of ARM’s share price from less than £1 in early 2009 to £9.80 at 10.44 GMT January 24 2014 reflects management PLT, their innovation and their attack on Intel’s dominance in the chip market and Intel’s lagged response to getting its chips into smartphones and tablets. Intel management were bounded rational. They tried to acquire ARM but antitrust law prohibited the acquisition. Tobin’s q is interchangeable with Marris v. Both rely on market valuations; the Marris v, however, should be understood in terms of PLT. Management’s type, that is, their ability to define the game dimension and their ability to win the game represent an intangible asset in the Marris v. The Marris v by relying on market valuation avoids many of the descriptors of accounting profits wherein high profits often equate with a monopoly position. But it could also be the case that companies with high market shares earn profits not attributable to concentration in the market – they are more efficient and more innovative than their competitors. Observe the share price and the investment commentary but when v < 1 or v < v step back, read the signals, make a judgement call and consider a buy as a long term investment – do not look back and do not regret the decision once made.

Memo to Ms Ahrendts


Re: Apple Inc: Play not to lose: Minimax strategy

Dear Ms Ahrendts

Congratulations on your recent appointment. We have been commenting on Apple for a number of years in this Blog, and from the perspective of game theory. You should challenge everything about the data – market share figures, consumer loyalty and the source of the competitive threat. Apple does need to refocus, to reshape its strategy in order to compete in an evolving game that exhibits both convergent technologies and rapidly changing set of consumer preferences. Are you a brand? Are you a design company or an innovator? Analysts look at Apple in terms of profit margins and a company trading on earnings estimates and revision of the estimates. With new product launches across the i-suite of products, coupled with an underlying iOS ecosystem, they look forward to new product launches, and endless queues by early adopters and loyal fans at different cities across the world. But from our perspective, observing Apple as a player in a game, we would adjudge that you are not winning the game.

Confused consumers

First of all, your product offerings are in danger of becoming nodoids: in other words, they come to represent nothing more than a roll-out across a common platform of a suite of not dissimilar products absent any innovation. Consumers are either underwhelmed or disappointed. Once they ask the nodoid question: ‘is an iPhone an iPad or is the iPad an iPhone?’ the game dynamic switches from a game of playing to win to a game of playing not to lose. This is happening. Secondly, the analysts expect the i-Watch – so what? Analysts continue to debate the next big thing. So what? Could it be IPTV or cloud solutions?  So what? You know that you are not in search, you know that you are not in digital mobile advertising, you are a late entrant into cloud services, you failed to acquire Twitter, SIRI failed, Newton failed in the 1990s and in 2013 you allow us to believe that you are not a player in IPTV.

We have argued this before #tuncnunc discussing a range of game solutions to consider: launch a nano iPhone or engage in a telecom alliance with 4G LTE providers such as China Mobile. The 5C launch is about maximising profit margins; a nano offensive play, however, would ignite a $99 ‘sweet price’ competition for full functionality smartphone devices. Forward guidance on the stock estimate above $500 may adjust for these events in 2014-15 but these events may now be too late from a game perspective to play to win the long game. In other words, no longer is it about how Apple is performing in 2013, it should be about Apple’s likely performance in 2023.



Second mover advantage: SMA & Minimax

So an alternative for you to consider in your new role is to secure the second mover advantage [SMA] by playing not to lose. First, recognise that your market shares are increasing at a decreasing rate. Correct that trend. The iPhone 5 delay, for example, created a zero-sum switch to rivals, notably Samsung, in the UK and possibly across the EU. Your smartphone market share is under threat in Asia as the convergent smartphone and tablet game evolves to become Apple’s game to lose. Start thinking like your competitors – reason like this: ‘I think-you think-I-think’: Apple thinks that Samsung expects it to defend the iPhone, so Samsung will attack the iPad. But Samsung believes that Apple will reason this way, and so assuming that Apple will defend the iPad, Samsung will attack the iPhone. But Samsung also knows that Apple will reason this way.

This line of reasoning suggests that some kind of a decision tree ‘what-if’ analysis will reveal which strategy is Apple’s optimal choice. But it is more complex than that – we argue in our new book Decoding Strategy that how either player does in the game depends on what each believes the other is likely to do. Apple has to choose to play a minimax strategy, that is, a strategy that minimizes the maximum amount Samsung can expect to get in the evolving smartphone and tablet game, and thus maximize the amount Apple can expect to win. It is for you to patch a minimax strategy into your strategic vision for 2014 and beyond. To quote T.S.Eliot: ‘What we call the beginning is often the end, and to make an end is to make a beginning, the end is what we start from’. With best wishes in t+1…..