5.5 Sequential Games

[5.2 Using Game Theory] [5.3 Classic Game Models] [5.4 Simultaneous Games]

What Are Sequential Games?

Sequential games are those in which players make moves at different times or in turn. This means that players who move later in the game have additional information about the actions of other players or states of the world. This also means that players who move first can often influence the game. Each player's strategy makes the actions that he or she chooses conditional on the additional information received during the game.

Understanding sequential games is very important in business. It is common for business planners to apply rule of thumb approaches and static analysis to situations. However, this approach ignores the fact that strategic situations are often vastly different from one another and very dynamic. Modelling business situations as sequential games forces a planner to consider these aspects and allows for better forecasting and planning, both of which lead directly to better decision-making.

Extensive Form Representation of Games

A common way of representing games, especially sequential games, is the extensive form representation, which uses game trees. Game trees are made up of nodes and branches, which are used to represent the sequence of moves and the available actions, respectively. Consider two players, Mr Black and Ms White, who are playing a sequential game. Mr Black moves first and has the option of Up or Down. Ms White then observes his action. Regardless of what Mr Black chooses, she then has the option of High or Low. The game tree for this game would appear as follows:

In this subject, decisions are represented by square nodes. Node a is the decision node where Mr Black chooses between Up and Down. Since node a is the first node, it is also known as the initial node. Nodes b and c are the decision nodes at which Ms White chooses between High and Low. The triangle-shaped ending nodes on the right are the terminal nodes, which also have the payoffs for each player associated with each outcome listed beside them.

Sometimes, one player's action at a given stage can change the options available at subsequent stages. Suppose that you adjust the above game so that if Mr Black plays Down, Ms White can play High, Medium and Low. In this situation, the game tree would look as follows:

Strategies in Sequential Games

In sequential games, it is important to clearly define what is meant by strategy. Game theorists define a strategy as a complete contingent plan of actions. In other words, a strategy specifies what action a player will take at each decision node. Consider once again the game between Mr Black and Ms White.

Mr Black has two strategies available — Up and Down. Ms White, however, actually has four strategies available since there are two nodes to consider — b and c — and two possible actions at each node — High and Low. The following table shows the strategies available to Ms White:

	If Mr Black Plays Up, Play:	If Mr Black Plays Down, Play:
Ms White's Strategy 1	High	High
Ms White's Strategy 2	High	Low
Ms White's Strategy 3	Low	High
Ms White's Strategy 4	Low	Low

Because actions always lead to reactions, an important aspect of strategy in sequential games is that players must consider — and plan for — their opponent's reactions. In the example above, if Mr Black wants to maximise his payoff, he must consider how Ms White will react if he moves Up and how she will react if he moves Down.

The following animation illustrates how game trees can be used to map out sequential games. As you watch, take special note of how the decisions of one player affect the strategy choices available to the other.

Decision and Game Trees

Decision and game trees are used to map out the types of scenarios businesses encounter every day. Decision trees are used to map out scenarios involving only one player; game trees are meant to handle scenarios with multiple players. In either case, it is important to remember that trees are simply tools — they do not take the decision out of the decision-maker's hands. Instead, they are intended to channel that person's experience and intuition toward the goal of finding the best strategy given the known alternatives. In the end, the decision-maker is left with a likely "probability" rather than a simple guess.

Every tree is based on certain assumptions. The goal is to limit these to the most relevant assumptions for the given scenario. In this way, the game tree tool remains useful and manageable. Otherwise, if every imaginable situation was included, evaluating the remote chance of, say, nuclear war, would get in the way of assessing the more relevant chance that a competitor has the resources to enter the market first.

The advantage of game trees

Game tree models allow players to make better decisions by forcing them to consider the actions and reactions of all other players involved. In a sequential game, the decision-maker eliminates a great deal of uncertainty simply by creating a clear-cut list of the various players, their actions and reactions, and the decision-maker's best response to each. The game tree provides a formal means to keep track of these items. Without this method, many players or alternatives might be overlooked and therefore never planned for. Lack of planning leads to surprises, and surprises from the competition are rarely enjoyable.

For example, suppose player 1 is considering whether to move Up or Down in its game with player 2 and creates the following game tree without considering player 2's payoffs.

With the current tree, player 1 has no way of knowing what player 2 will do at node b. Player 2 has an equal chance of choosing Up (giving player 1 a payoff of $500) or choosing Down (giving player 1 only $100). Based on the available information, player 1 has no choice but to assume that he or she has no better than a 50/50 chance of getting $500 by choosing Up, but is guaranteed to get $450 by choosing Down. So while player 1 might want the $500, he or she will optimally choose Down and take the $450 to avoid the risk.

Without taking an opponent's payoffs or motivations into account, a player cannot be confident in making a decision. The tree below features player 2's payoffs. Is Down still an optimal choice for player 1 now that player 2's payoffs have been considered?

The answer is no. When player 1 takes player 2's reactions into account, player 1 sees that Up is the optimal move at node a because player 2 would also optimally choose Up at node b (where player 2 will earn $300 rather than only $200 from Down). By considering all the elements of this game, player 1 ends up with the $500 originally hoped for.

Chance Nodes

In addition to decision nodes, a game tree can include chance nodes (represented in this subject by circles). Chance nodes are used to symbolise events that are uncertain or beyond a primary player's direct control.

Suppose you were standing at the foot of the Himalayas and had the option of either trying to climb Mt. Everest or walking away. In a decision or game tree, that choice (Climb or Don't Climb) would be represented by a decision node because it is entirely in the hands of you, the player — only you can decide whether you will try to climb the mountain.

On the other hand, how far up the mountain you will get is largely uncertain or "up to chance". You can aspire to reach the summit, but you cannot merely decide to get there — weather, injury, lack of sufficient supplies, etc, could all end your climb early. Therefore, the question of whether your climb will succeed or fail is represented by a chance node (indicated by the letter C), as in the following tree.

Chance Players

Every game is built around the actions and payoffs of the primary players. Chance players are people or organisations that have the potential to influence the game, even though these players are not directly competing for the game's payoffs. When constructing a game tree, chance nodes are used to represent events (pure chance) or players that are out of the direct control of the primary players but that, nonetheless, must still be factored into the decisions made by the primary players.

In the case of the competing researchers, a government agency interested in possibly giving a research grant to one of the competing companies would be considered a chance player. This is because the primary players cannot decide whether (or to whom) the agency will actually give the grant. It is out of the hands of the primary players.

The important thing to remember is that primary players — those competing directly for that game's payoffs — have no way of forcing chance players to act in a desired or beneficial manner. By definition, chance players are not affected or directly influenced by the actions of the primary players.

Returning to the example of climbing Mt. Everest, a chance player, in this case, might be an experienced Sherpa guide who has the choice of leading your expedition or that of another climber. You and the other climber would want the Sherpa's help, but neither of you can control the decision. It is entirely up to the Sherpa to decide which expedition to lead. The following tree illustrates the way chance players and pure chance (C) work in the context of decision or game trees:

As with assumptions, it is important to limit chance events to those most relevant to the scenario the decision-maker is analysing.

Click on the links below to see two examples that further illustrate how game trees can be used.

To Discount or Not to Discount

After a scenario is mapped out in a game tree model and the different possibilities resulting from the different options are clearly laid out, a decision-maker will still not be able to choose the best strategy. The next step after creating the tree is to "solve" it — to begin to strip away unlikely branches until only the dominant strategy remains. This is done through backward induction.

Backward Induction

In this process, the game tree is essentially flipped. Working backwards from the payoff, the decision-maker begins to eliminate suboptimal actions until only the most likely path remains. By doing this, an opponent's likely moves from the initial node to the payoff can be mapped, allowing the decision-maker to strategize for each of those potential moves and ultimately find equilibrium.

Backward induction assumes that players will move optimally at each node — that opponents can be expected to act in their own best interests. Knowing this, a decision-maker working to solve a tree can confidently eliminate actions that are suboptimal to his or her opponents. For example, consider the earlier game between Mr Black and Ms White.

At node b, playing High gives Ms White a payoff of 0, while playing Low gives her a payoff of 2. Therefore, Ms White would rationally choose to play Low. We can ignore the possibility of Ms White's playing High at node b. Similarly, we can ignore the possibility that Ms White will play Low at node c since her payoff for High is 1 and for Low is 0. In short, of the four strategies available to Ms White, backward induction implies that her only rational strategy is to play Low at node b and High at node c. This implies that Mr Black's choices look as follows:

Notice that Mr Black's optimal strategy is now obvious — play Down. Down yields a payoff of 2 while Up gives a payoff of only 1. Consequently, the equilibrium of this game is Mr Black playing Down and Ms White playing Low if Mr Black plays Up and playing High if Mr Black plays Down. This is called the subgame perfect equilibrium of the game.

Nash Equilibrium versus Subgame Perfect Equilibrium

It is important to note that all subgame perfect equilibria are Nash equilibria. Since backward induction ensures that each player will play his or her best action at each node, the resulting strategies will correspond to a Nash equilibrium. To see this, again consider the game between Mr Black and Ms White.

Notice that there is only one subgame perfect equilibrium. Ms White plays Low if Mr Black plays Up, and plays High if Mr Black plays Down. Therefore, Mr Black will play Down.

However, suppose that Ms White has adopted a strategy that states that she should always play Low and Mr Black has chosen to play Up. Can a Nash equilibrium be reached in this case?

A Nash equilibrium implies that no player can do better by switching strategy given the strategies of the other players. If Ms White switches her action to High at node b but still chose Low at node c, then she would be worse off given that Mr Black is playing Up. If she switches her action to high at node c, continuing to choose Low at b, then she would be no better off given that Mr Black already is playing Up. Similarly, with Ms White committed to always playing Low, if Mr Black chose to switch his strategy to play Down, he would be worse off. Thus, no player can unilaterally be better off by switching his or her strategy.

We have shown that this result is a Nash equilibrium, but it is not a subgame perfect equilibrium. This is because it violates the rules of backward induction, which hold that Ms White would never choose Low at node c. In summary, all subgame perfect equilibria are Nash equilibrium, but not all Nash equilibrium are subgame perfect equilibria.

You can also determine whether Ms White always playing Low and Mr Black playing Up is a Nash equilibrium by using the following payoff matrix:

Payoff Matrix		Mr Black
Payoff Matrix		Up	Down
Ms White	High/High	0, 0	2, 1
	High/Low	0, 0	0, 0
	Low/High	1, 2	2, 1
	Low/Low	1, 2	0, 0

Note: Ms White's choices specify her actions at node b and node c, respectively.

Some games include both sequential and simultaneous elements. For example, a game might initially consist of an entry decision by one firm. If the firm enters, there will then be a simultaneous competition game. The game would be solved by backward induction as well; in the last stage, you can solve for the Nash equilibrium and work back in the tree, assuming the payoffs from the Nash equilibrium in the last stage.

First- or second-mover advantage

In the game between Mr Black and Ms White, Mr Black was able to use a first-mover advantage to achieve the outcome that he preferred. Many times, by moving first, a player can determine the direction of the game — forcing other players to then react to that choice rather than moving on independently. However, not all sequential games have a first-mover advantage. In fact, some have a second-mover advantage. For example, consider the following game:

If Amy chooses Up, Bernard will optimally choose Zag. If Amy chooses Down, Bernard will optimally choose Zig. In both cases, Bernard will end up with a payoff of 1 while Amy will have a payoff of –1. However, if we switch the order so that Bernard moves first and Amy moves second, then Amy will end up with a payoff of 1 while Bernard will have a payoff of –1. Therefore, this type of game has a second-mover advantage.

Do first movers always have an advantage? Click on the following here to see why moving first might be a disadvantage.

An important facet of sequential games is that players would often like to arrive at a particular Nash equilibrium but cannot because it is not a subgame perfect equilibrium. Interestingly, this is often the player's own fault. Consider the game between Mr Black and Ms White again.

Ms White prefers Outcome X to Outcome Y, but she cannot get there unless Mr Black plays Up. She needs some method to induce Mr Black to play Up. Notice that Mr Black would prefer to play Up if Ms White always played Low. This would result in Outcome X.

You might wonder why Ms White cannot simply threaten to always play Low, and thereby, achieve Outcome X. This is because the threat is a . Mr Black knows that once he plays Down, Ms White will choose High regardless of the threats that she made. Playing High is simply Ms White's best move at node c.

This demonstrates an important concept in game theory — the value of commitment. If Ms White could credibly commit to always play Low, then Mr Black would choose to play Up and Outcome X would result.

Click on the following link to see how one company's commitment was beneficial to keeping rivals out of the market.

How Commitment Helps

Commitment limits the options available to a player. People often make a mistake by assuming that more options are better. However, game theory presents several situations in which more options make players worse off. Therefore, players often find it in their best interests to limit their available actions in advance, such as when Cortes burned his ships to eliminate retreat as an option.

Unfortunately, though, eliminating options in a manner as physical and permanent as the burning of a ship is rarely possible in business. However, if a player is able to commit to a course of action in such a way that all other players recognise that the committed player will never do the opposite, that player has effectively eliminated that undesirable option.