Payoff Table and Tree Part1

ACCA F5 - Performance Management 

Payoff table:

(A) Action/Alternative = Best known as alternative course of action or the available choices for decision maker. For each choices decision maker can exert their discretion. E.g. In a summer month an ice-cream seller can order large, medium, or small batch of ice cream knowing that the sales will be lower if days are cloudy and higher if sunny. Ordering is the action. Alternative are large, medium and small batches and is under control of ice-cream seller.

(E) Events = Factors which are not in control of decision makers. The event is not under control of decision maker. E.g. In above case the rain is the factor, which determines the sales volume for ice-cream seller. If the day is cloudy, only few people prefer ice cream while if the day is sunny it is likely that ice-cream demand will be high. Therefore, factor which triggers demand is type of day and is not under control of the decision maker.

(P) Probability: The effect of action by likelihood of factor is measured by probability. Probability is outcome of an action assigned to an event. E.g. 95% of sales for small batch if the day is cloudy or 100% if the day is sunny.

Construction a probability table: Remember A\E

ACTION        \                  EVENT	Cloudy	Sunny
Small batch	95%	100%
Medium batch	75%	100%
Large batch	40%	90%

(P) Payoffs = Payoff is the outcome of a particular action for a particular event. Outcome is the decision point. In business payoff is expressed in terms of profits, costs, net present value, required rate, or return on investment. E.g. Profit is the outcome for ice-cream seller and each profit level for different orders and sales are payoffs. E.g. the profit figure is the outcome for ice-cream seller. We know profit is the extra amount received on sales, which exceed costs.

Constructing payoff table: Selling price per unit of ice-cream is $15 and unsold ice-cream is send back to supplier for which half the cost is refunded.

ACTION        \     EVENT	Cloudy			Sunny
	Sales	Loss on return	Total	Sales	Loss on return	Total
Small batch 50 units costing $10 each with 2 varieties	95%*50*(15-10) = 237.5	5%*50*(10*50%) = (12.5)	225	100%*50*(15-10) = 250	0	250
Medium batch 100 units costing $9 each with 3 varieties	75%*100*(15-9) = 450	25%*100*(9*50%) = (112.5)	337.5	100%*100*(15-9) = 600	0	600
Large batch 200 units costing $8 each with 4 varieties	40%*200*(15-8) = 560	60%*200*(8*50%) = (480)	80	90%*200*(15-8) =1260	10%*200*(8*50%) = (80)	1180

Summarizing payoff table:

ACTION        \                  EVENT	Cloudy	Sunny
Small batch	225	250
Medium batch	337.5	600
Large batch	80	1180

Maxi-max

ACTION        \     EVENT	Cloudy	Sunny	Maximum
Small batch	225	250	250
Medium batch	337.5	600	600
Large batch	80	1180	1180
Maximum value from the maximum column (Maxi-max) is for large batch			1180

Maxi-min

ACTION        \     EVENT	Cloudy	Sunny	Minimum
Small batch	225	250	225
Medium batch	337.5	600	337.5
Large batch	80	1180	80
Maximum value from the maximum column (Maxi-min) is for medium batch			337.5

Regret (Opportunity loss): Opportunity is described as the chance to maximization of payoff for an event. Thus, the opportunity loss (regret) is deviation of each payoff for an event from maximum payoff of that event.

ACTION        \                  EVENT	Cloudy	Sunny
Small batch	337.5 - 225 = 112.5	1180 - 250 = 930
Medium batch	337.5 - 337.5 = 0	1180 - 600 = 580
Large batch	337.5 - 80 = 257.5	1180 - 1180 = 0

Min-max Regret:

ACTION        \     EVENT	Cloudy	Sunny	Maximum
Small batch	112.5	930	930
Medium batch	0	580	580
Large batch	257.5	0	257.5
Minimum value from the maximum column (Mini-max) is for large batch			257.5

Payoff tree:

(A) Action/Alternative = Best known as alternative course of action or the available choices for decision maker. For each choices decision maker can exert their discretion. E.g. In a summer month an ice-cream seller can order large, medium, or small batch of ice cream knowing that the sales will be lower if days are cloudy and higher if sunny. Ordering is the action. Alternative are large, medium and small batches and is under control of ice-cream seller.

(E) Events = Factors which are not in control of decision makers. The event is not under control of decision maker. E.g. In above case the rain is the factor, which determines the sales volume for ice-cream seller. If the day is cloudy, only few people prefer ice cream while if the day is sunny it is likely that ice-cream demand will be high. Therefore, factor which triggers demand is type of day and is not under control of the decision maker.

(P) Probability = The effect of action by likelihood of factor is measured by probability. Probability is outcome of an action assigned to an event. E.g. 95% of sales for small batch if the day is cloudy or 100% if the day is sunny.

Construction a probability tree: Remember A\E                 

Decision Point	Alternatives	Chance Point	Events	Happening Point	Outcome	Quantifying Outcome
	Small batch		Cloudy		Sales	95%
					Return	5%
			Sunny		Sales	100%
					Return	0%
	Medium batch		Cloudy		Sales	75%
					Return	25%
			Sunny		Sales	100%
					Return	0%
	Large batch		Cloudy		Sales	40%
					Return	60%
			Sunny		Sales	90%
					Return	10%

(P) Payoffs = Payoff is the outcome of a particular action for a particular event. Outcome is the decision point. In business payoff is expressed in terms of profits, costs, net present value, required rate, or return on investment. E.g. Profit is the outcome for ice-cream seller and each profit level for different orders and sales are payoffs. E.g. the profit figure is the outcome for ice-cream seller. We know profit is the extra amount received on sales, which exceed costs.

Constructing payoff tree: Selling price per unit of ice-cream is $15 and unsold ice-cream is send back to supplier for which half the cost is refunded.

Decision Point	Alternatives	Event  Point	Events	Happening Point	Outcome	Quantifying Outcome	Payoff
	Small batch		Cloudy		Sales	95%*50*(15-10) = 237.5	=273.5-12.5 =225
					Return	5%*50*(10*50%) = (12.5)
			Sunny		Sales	100%*50*(15-10) = 250	=250
					Return	0
	Medium batch		Cloudy		Sales	75%*100*(15-9) = 450	=450-112.5 =337.5
					Return	25%*100*(9*50%) = (112.5)
			Sunny		Sales	100%*100*(15-9) = 600	=600
					Return	0
	Large batch		Cloudy		Sales	40%*200*(15-8) = 560	=560-480 =80
					Return	60%*200*(8*50%) = (480)
			Sunny		Sales	90%*200*(15-8) =1260	=1260-80 =1180
					Return	10%*200*(8*50%) = (80)

Summarizing payoff tree:

Alternatives	Events	Payoff
Small batch	Cloudy	225
	Sunny	250
Medium Batch	Cloudy	337.5
	Sunny	600
Large Batch	Cloudy	80
	Sunny	1180

Maxi-max

Alternatives	Events	Payoff	Maximum
Small batch	Cloudy	225	250
	Sunny	250
Medium Batch	Cloudy	337.5	600
	Sunny	600
Large Batch	Cloudy	80	1180
	Sunny	1180
Maximum value from the maximum column (Maxi-max) is for large batch			1180

Maxi-min

Alternatives	Events	Payoff	Minimum
Small batch	Cloudy	225	225
	Sunny	250
Medium Batch	Cloudy	337.5	337.5
	Sunny	600
Large Batch	Cloudy	80	80
	Sunny	1180
Maximum value from the maximum column (Maxi-min) is for medium batch			337.5

Regret (Opportunity loss): Opportunity is described as the chance to maximization of payoff for an event. Thus, the opportunity loss (regret) is deviation of each payoff for an event from maximum payoff of that event.

Alternatives	Events	Payoff (X)	Best Payoff for Events (Y)	Regret (Y-X)
Small batch	Cloudy	225	337.5	112.5
	Sunny	250	1180	930
Medium Batch	Cloudy	337.5	337.5	0
	Sunny	600	1180	580
Large Batch	Cloudy	80	337.5	257.5
	Sunny	1180	1180	0

Min-max Regret:

Alternatives	Events	Regret	Minimum
Small batch	Cloudy	112.5	930
	Sunny	930
Medium Batch	Cloudy	0	580
	Sunny	580
Large Batch	Cloudy	257.5	257
	Sunny	0
Minimum value from the maximum column (Mini-max) is for large batch			257.5