Game Theory in Life Essay

Game theory is a proposition used for strategic decisions. Mathematical models are used to examine the interaction between the decision makers, the issue at hand, and the cooperation between the decision makers.

Case Study

The conflict issue is about Iran and Israel. A rivalry for influence and power is emerging between the two nations. Iran is becoming uncomfortable with the regional competition posed by Israel. Iran views such competitionas an intention to undermine her revolutionary system.

Israel, on the other hand, is disturbed by Iran’s anti-Jewish ideologies together with her increasing military potential, particularly her nuclear energy programs (United Institute of Peace 1). Israel believes that Iran’s nuclear program will be used to create nuclear weapons, which Iran might use to wipe Israel from the world map.

Israel already has nuclear energy programs in place. She is worried about Iran’s intentions and is not interested much in offending Iran unless she believes that Iran has crossed the red line, meaning that she is ready to engage in a nuclear war. Iran is also focused on avoiding to show early offence to Israel since she might be resented by a large section of the international community, or even face combined strikes from the UN and Israel allies.

It has demonstrated this by holding talks with the International Atomic Energy Agency (IAEA) about inquiries into her atomic bomb research (Thomson Reuters 2). Israel, however, is still convinced that Iran is being cleverly deceptive, and therefore wants to engage her directly. The two countries have to respond to the tension that has built between them, and the results of their response might be beneficial or fatal to both or one of them.

Summary of the game

Issue: Does either of the countries need to boost her military capability in anticipation of war, or embrace more peaceful strategies in order to facilitate arbitration?
The players are Israel and Iran. The actions are boosting military capacity to attack the rival, and receding from war preparations in order to maintain peace.

The payoffs for the game include:

• A nation loses resources by fighting an unprepared rival and gets a payoff of (-1). Remember that winning the war does not solve the conflict.
• The unprepared nation falls if attacked by a well-prepared rival. The defeated nation gets a payoff of (-3).
• A well-prepared nation fights another well-prepared nation, and the battle ends in a draw, and apayoff of 2is gained by each nation. In this case, both nations defend their resources against the rival’s strategies, say each nation fails to strike well because the other one has ensured perfect selfprotection against attack.
• Both nations enjoy peace when Iran makes no weapons and Israel trusts her (each of the nations get a payoff of 3).

Embracing peace can work for the benefit of both nations if Iran acts in the interest of peace by not manufacturing nuclear weapons, and Israel trusts Iran. Iran can do this by subduing her nuclear capacities and only arming her military to a level that is normal (for defensive purposes only). Israel, on the other hand, will have to respond optimistically by maintaining peace.

Sets of strategies

This equilibrium is inherently stable, i.e. what Israel is doing is optimal given what Iran is doing and vice versa and thus none of them would regret her move.

3, 3 is the oneset of stable strategies, and is the first Nash equilibrium. In this set, if both countries keep peace and limit their military capacities to just normal (for defensive purpose only), then neither of them would wish to strike because such a move is unprofitable because it gives a payoff of (-1).

2, 2 is also a stable set and is the second Nash equilibrium.If both nations beef up their military capacities to levels that escalate conflict and result in war, then none of them would wish to recede. Receding gives a payoff of (-3), which is the worst payoff in the game.

Neither of the nations would want to fall whether in the presence or absence of war. In a Nash equilibrium, all players follow some rule or natural law, in this case, the law is to survive whether in presence or absence of war, without focusing on any incentive (Boleslavsky, “Mixed Strategy Nash Equilibrium” 6).

A best responseis a strategy a where player cannot gain more utility by switching to another strategy.Best responses for Israel are:

• to keep peace if Iran keeps peace, and gain a payoff of 3 instead of fightingher for a payoff of (-1).
• to fight back if Iran escalates the conflict to a war,and get a payoff of (2), instead of (-3) obtained from receding.

Israel’s best responses are marked by asterisks in the table below:

Best responses for Iran areto keep peace and get a 3 if Israel keeps peace, instead of getting (-1) by fighting, and to fight back and get a 2 if Israel escalates conflict into a war, and avoid a payoff of (-3) that would be gained from keeping peace. Iran’s best responses are marked by asterisks in the table below:

In the two sets of equilibrium, no player has an incentive to change her strategy, each has to respond to the other’s actions in a manner that would optimize her utility and therefore:

• Only individual deviations are useful.
• There are no useful group deviations.

Due to the different weights placed on various strategies, an algorithm for a mixed strategy Nash equilibrium has to be developed. It will allow us to find the mixed strategies available to each of players, which are meant to make their rival indifferent in her choices (Boleslavsky, “Dynamic Games” 5).

Mixed strategies in this case are strategies that:

• Israel can use to make Iran indifferent on whether to embrace peace or fuel the conflict.
• Iran can use to make Israel indifferent on whether to embrace peace or fuel the conflict.

To come up with this algorithm:

• Let P1 stand for move towards embracing peace by Iran, and P2 stand for move towards embracing peace by Israel.
• Let W1 stand for move towards escalating conflict by Iran,and W2 stand for move towards escalating conflict by Israel.

Whenever Israel choses to embrace peace, i.e. avoid anticipating for war by trustingIran, and avoid launching attacks, Iran can chose a randomization strategy to either amass weapons secretly or embrace peace by keeping her military capacity at defensive level only.

If Iran choses to embrace peace, i.e. tame her behavior in nuclear program, and keep her military capacity at defensive level, Israel may randomize her choices between trusting Iran so as to embrace peace, and launching war regardless of Iran’s promises to tame herbehavior in the nuclear program.

Each country’s mixed strategy is aimed at making the other one’s payoffs (expected utilities) averagely the same, whether they chose to escalate the conflict or embrace peace.

Israel’s Mixed Strategy

Israel’s mixed strategy is one that makes Iran’s Expected Utility (EU)for choosing to escalate conflict as a pure strategy, equal to that gained from embracing peace as a pure strategy. This can be represented as:

EU_P1 = EU_R1

Iran’s expected utility for choosing peace is a function of a mixed strategy where Israel choses peace as shown below:

EU_P1 = f (σ_U)

The same applies if Iran choses to escalate the conflict as shown:

EU_w1 = f (σ_U)

Israel’s mixed strategy can be solved by examiningthe expected utility for Iran’s move to limit military capacity and be honest, as a function of mixed strategy for Israel’smove to trust Iran, and embrace peace.

For Israel’s mixed strategy

Some percentage of the time, Iran is getting a 3 when Israel moves towards peace, and a (-1) the rest of the time when Israel plays towards causing war. Iran’s expected utility for playing towards peace is:

EU_P1 = σ_U(3) + (1 – σ_U)(-1), whereby (1 – σ_U) is the percentage of this time when Israel moves towards war.

Similarly, Iran’s expected utility for choosing conflict is EU_W1 = σ_U(-3) + (1 – σ_U) (2)

Since EU_P1 = EU_w1,yet we haveEU_P1 = σ_U (3) + (1 – σ_U) (-1), and EU_W1 = σ_U (-3) + (1 – σ_U) (2):

σ_U (3) + (1 – σ_U) (-1) = σ_U (-3) + (1 – σ_U) (2)

By simplifying this, 3σ_U – 1 + σ_U = -3σ_U + 2 – 2σ_U

3σ_U + σ_U + 3σ_U + 2σ_U =2 + 1, therefore 9σ_U =3

σ_U =1/3 meaning that if Israel plays towards peace 1/3^rd of the time, and towards causing war 2/3^rd of the time, then Iran is indifferent on whether to escalate the conflict or tame her nuclear activities and avoid war.

Iran’s Mixed Strategy

When the same method is applied for Iran’s mixed strategies;

EU_P2 = EU_W2

EU_P2 = f (σ_W1)

The same applies if Iran choses to escalate the conflict as shown:

EU_W2 = f (σ_W1)

EU_P2 = EU_W2yet we haveEU_P2 = σ_W1 (3) + (1 – σ_W1) (-1), and EU_W2 = σ_W1 (-3) + (1 – σ₁) (2):

σ_W1 (3) + (1 – σ_W1) (-1) = σ_W1 (-3) + (1 – σ_W1) (2)

By simplifying this, 3σ_W1– 1 + σ_W1= -3σ_W1+ 2 + 2σ_W1

3σ_W1 + σ_W1+ 3σ_W1+ 2σ_W1=2 + 1

σ_W1=1/3

This result means that if Iran plays towards peace 1/3^rd of the time and towards war 2/3^rdof the time, then Israel is indifferent on whether to escalate the conflict or attack her, and cause war. As a result of this outcome, the optimal strategy for the nationsis for both Israel and Iran to play towards peace with probability 1/3, and away from it with probability 2/3, so that none of them can change her strategy in a way that can give her competitive advantage over her rival.

Works Cited

Boleslavsky, Raphael. Dynamic Games 1 (2011): 5.

Boleslavsky, Raphael. Mixed Strategy Nash Equilibrium. 1 (2011): 6.

Thomson Reuters. “U.N. Nuclear Agency in Talks about Talks with Iran.”The Reuters. Mon Apr 22, 2013: 1-3. Print.

United Institute of Peace. “Israel and Iran: A Dangerous Rivalry from Iran Primer.”The Iran Primer.January 10 (2012):1. Print.