APL (Dyalog Unicode), 4 bytes^SBCS

Anonymous tacit prefix function. Takes [a,b,c,d] as argument. Returns a bit-Boolean array.*

⌈/=⌽

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⌈/ Does the maximum of the argument

= equal (vectorises)

⌽ the reverse of the argument?

* Note that APL stores arrays of Booleans using one bit per value, so this does indeed return a 4-bit word, despite the display form being 0 0 1 0.

R, 33 bytes

sum(2^which(max(a<-scan())==a)/2)

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Using numerical output. if a vector of booleans is acceptable, then:

R, 17 bytes

max(a<-scan())==a

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JavaScript (ES6), 40 bytes

Takes input as ([d,c,b,a]).

a=>a.map(r=x=>r=r*2|x==Math.max(...a))|r

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Or 30 bytes if we can return 4 Boolean values:

a=>a.map(x=>x==Math.max(...a))

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Perl 6, 12 bytes

{$_ X==.max}

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Anonymous code block that takes a list of integers and returns a list of booleans. If we need to return as a number, it's +4 bytes to wrap the inside of the code block with 2:[...].

Explanation:

{          }  # Anonymous code block

 $_           # With the input

    X==       # Which values are equal

       .max   # To the maximum element

Japt, 10

£Â(X¶UrÈwY

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-1 byte thanks to @Oliver!

Input is a 4-element array in the following format:

[d, c, b, a]

Output is an array of bits.

05AB1E, 3 bytes

ZQR

Input as a list of [a,b,c,d], output as a list of boolean.

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If this output format is not allowed based on the hexadecimal that is used in the challenge description, it would be 6 bytes instead:

ZQRJCh

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Explanation:

Z    # Take the maximum of the implicit input-list

 Q   # Check for each in the (implicit) input-list if it equals this value

  R  # Reverse the resulting list (and output implicitly as result)



J    # Join the list of 0s and 1s to a single string

 C   # Convert from binary to integer

  h  # Convert from integer to hexadecimal (and output implicitly as result)

Java (JDK), 85 bytes

a->{int m=a[0],w=1,i=0;for(;++i<4;)w=a[i]<m?w:(1<<i)^(a[i]>m?(m=a[i])^m:w);return w;}

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• Takes the input as an array of [a,b,c,d].

Haskell, 47 bytes

a#b=sum[1|b]+2*a

foldl(#)0.(map=<<(==).maximum)

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35 bytes if I can output a string

f x=show.fromEnum.(==maximum x)=<<x

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PHP, 54 bytes

while($i<4)$argv[++$i]<max($argv)||$r+=1<<$i-1;echo$r;

or

while($i<4)$argv[++$i]<max($argv)||$r+=1<<$i;echo$r/2;

take input from command line arguments. Run with -nr or try them online.

C# (Visual C# Interactive Compiler), 39 bytes

n=>n.Select((a,b)=>n[3-b]==n.Max()?1:0)

Returns an IEnumerable<int>, which represent bits.

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C# (Visual C# Interactive Compiler), 35 bytes

n=>n.Select((a,b)=>n[3-b]==n.Max())

If an IEnumerable<bool> is acceptable.

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C# (Visual C# Interactive Compiler), 49 bytes

n=>{for(int i=4;i-->0;Write(n[i]==n.Max()?1:0));}

Prints a binary string to STDOUT.

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Here's a JS version that outputs as binary

update: Shorter with join, and without the lookup:

JavaScript (Node.js), 42 bytes

a=>a.map(x=>+(x==Math.max(...a))).join('')

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Previous, with lookup, 49 bytes

a=>a.map(x=>[0,1][+(x==Math.max(...a))]).join('')

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Previous, with reduce, 52 bytes:

a=>a.reduce((y,x)=>y+[0,1][+(x==Math.max(...a))],'')

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fa=>a.map(x=>+(x==Math.max(...a))).join('')

console.log(f([ 4, 1,77, 6])) // 0010

console.log(f([10,10, 5, 4])) // 1100

console.log(f([ 1, 1, 1, 1])) // 1111

C# (Visual C# Interactive Compiler), 51 bytes

x=>{for(int m=x.Max(),i=4;i-->0;)x[i]=x[i]==m?1:0;}

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Above is an anonymous function that outputs by modifying an argument. Output is an array of 1's and 0's.

Below is a recursive function that outputs an integer.

C# (Visual C# Interactive Compiler), 60 bytes

int f(intx,int i=3)=>i<0?0:2*f(x,i-1)|(x[i]==x.Max()?1:0);

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Both functions take input as a 4-element array.

[d, c, b, a]