In my last post, I introduced you to a new form of game called Sequential Games. Today, I’m going to explore with you a historical application of Sequential Game, the Battle of Hastings during the Norman Conquest.
The Norman Conquest refers to the invasion of Anglo-Saxon England led by William the Conqueror in 1066 with the hope of obtaining the English throne for himself. It is a major turning point in both English and European history. Now, we are going to look closer into William’s first and arguably the most important battle against King Harold, the Battle of Hastings.
In 1066, William the Conquer brought his army of around 7,000 soldiers across the English Channel from Normandy to Pevensey Bay in Sussex to attack the Saxons. At this time, King Harold was up North fighting another contender to the throne Harald Hardrada.
As far as William the Conquer knew, the Saxons had two choices: to fight or to flee. In each situation, William would then make the choice of fighting or fleeing. If the two armies meet and fight, given their relatively equal strength, neither of them will gain any advantage. If one of the armies flees and the other attacks, the attacking side gains 2 units of utility for gaining land without losing troop while the fleeing side gets 1 utility for preserving their force. If both parties flee, the Saxons will gain 2 utilities for preserving their land and force, while the Normans gain only 1 utility for preserving their army. We can present the two players decisions with the following decision tree:
Through backward induction, the Saxons knew that if they attack, the Normans would flee and the Saxons would gain 2 utilities, but if they flee, the Normans would attack and the Saxons would only gain 1 utility. Thus, the Saxons choose to attack. Thus, the outcome of this sequential game would be the Saxons attack and the Normans flee. This is obviously not a optimal situation for William the Conquer because he did not intend to bring his army across the Channel for nothing. Thus, he employed a bold tactic: burning all the boats after landing. This move eliminated some of his strategies in the game and completely changed the outcome. The situation after burning the ship can be represented by the following decision tree:
In this situation, the Saxons knowing that the Normans have no other option then to fight, would choose to flee. By burning his boats, William the Conquer secured a better payoff for his army by influencing his opponents strategy. In Game Theory, this is called Commitment Strategy, a player cutting off some of his strategies publicly to make his/her threats more credible. The key to a commitment strategy is that this player has to let his/her opponent KNOW of the plan so they will change their strategy according to the changing payoff. In this case, William the Conquer burnt his ship instead of drilling holes in them so that the Saxons knew they were not going back.
William’s tactic works perfectly in theory, but people who are educated on this topic will point out that the Saxons led by King Harold did attack William the Conquer anyway because they needed to defend their own territory. The Saxons were only defeated because King Harold was killed, according to Bayeux Tapestry, by an arrow in the eye. In our simplified model, patriotic feeling and other human emotions in this game are not accounted for. This historical event is once again an example of the limits of game theory models. It is a perfect reminder that we constantly need to adapt and improve our models according to specific situations.